Year 5 · Chapter 2

Year 5 Column Addition and Subtraction: worked examples and questions with answers

By the end of this chapter you will be able to

Add numbers with up to 6 digits using the column method, including carrying, zeros and numbers of different lengths
Subtract numbers with up to 6 digits using the column method, including exchanging across zeros
Add more than two numbers in columns
Add and subtract in your head with big numbers by partitioning, bridging, compensating, using number facts and near doubles
Round to estimate and to check answers, and choose how accurate an answer needs to be
Use the inverse operation to check, and to find missing numbers and missing digits
Solve multi-step problems with bar models, tables, charts, timetables and bank statements, and explain which method to use

You should already know

Adding and subtracting numbers with up to 4 digits in columns
Mental strategies: bridging, compensating and near doubles
Estimating by rounding to the nearest 10, 100 and 1,000
Bar models for part-whole and comparison problems
Place value to 1,000,000 (Chapter 1)

Where this leads

Multiplication (Chapter 4) and division (Chapter 5)
Decimals and money (Chapter 7)
Statistics: tables and line graphs (Chapter 11)
Order of operations and long multiplication in Year 6

National Curriculum for England, Key Stage 2 mathematics, Year 5. Content: Add and subtract whole numbers with more than 4 digits, including using the formal written methods of columnar addition and subtraction, Add and subtract numbers mentally with increasingly large numbers, Use rounding to check answers to calculations and decide, in the context of a problem, what level of accuracy is needed, Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why.

Printable worksheets. Every lesson in this chapter has a free worksheet and an answer sheet to download and print.

2.1 Column addition with bigger numbers: Worksheet (PDF) · Answers (PDF) · lesson page
2.2 Column subtraction with bigger numbers: Worksheet (PDF) · Answers (PDF) · lesson page
2.3 Adding and subtracting in your head: Worksheet (PDF) · Answers (PDF) · lesson page
2.4 Estimating, rounding and checking: Worksheet (PDF) · Answers (PDF) · lesson page
2.5 Bar models and multi-step problems: Worksheet (PDF) · Answers (PDF) · lesson page
2.6 Missing digits and number puzzles: Worksheet (PDF) · Answers (PDF) · lesson page
2.7 Tables, charts, timetables and bank statements: Worksheet (PDF) · Answers (PDF) · lesson page
Check-up and SATs-style questions: Worksheet (PDF) · Answers (PDF) · lesson page

Exercise P Do you remember?

Non-calculator

FWork out 4,386 + 2,547.

Done

FWork out 6,003 − 2,748.

Key words. Column subtraction is a written method. You write the bigger number on top and take away one column at a time. If the top digit is too small, you exchange: swap 1 of the next column for 10 of this column. The difference is the answer to a subtraction.

Method. Write the bigger number on top. Take away the ones first, then the tens, hundreds and so on. If a top digit is too small, exchange 1 from the next column.

Line up the digits in columns. The bigger number goes on top. Ones go under ones, tens under tens, hundreds under hundreds and so on.
Th H T O
6 0 0 3
− 2 7 4 8
Exchange, then subtract the ones. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
3 ones is not enough to take away 8. We need to exchange.
The tens and hundreds have 0, so there is nothing to exchange from there. We start further to the left, at the thousands.
Exchange 1 thousand for 10 hundreds: 6 thousands become 5 thousands, and 0 hundreds become 10 hundreds.
Exchange 1 hundred for 10 tens: 10 hundreds become 9 hundreds, and 0 tens become 10 tens.
Exchange 1 ten for 10 ones: 10 tens become 9 tens, and 3 ones become 13 ones.
13 − 8 = 5
Subtract the tens. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 4 = 5
Subtract the hundreds. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 7 = 2
Subtract the thousands. Now the next column to the left.
(This digit is 5 now, not 6, because we exchanged from this column.)
5 − 2 = 3
Read the answer. Check that the answer has a sensible number of digits, then write it.
Th H T O
56 90 90 133
− 2 7 4 8
3 2 5 5

6,003 − 2,748 = 3,255

Check. Use the opposite (inverse) operation: 3,255 + 2,748 = 6,003. ✓
Estimate: 6,000 − 2,700 = 3,300 (rounding to the nearest 100). It is close to the exact answer. ✓

Watch out. You cannot exchange from a 0. Look left for the first column that is not 0, then pass the exchange along one column at a time. Each 0 in the middle becomes 9.

Done

FRound 7,468 to the nearest 100.

Done

FWork out 546 + 99 in your head.

Done

FA school has 5,000 stickers. It uses 1,850 stickers on Monday and 2,375 stickers on Tuesday. How many stickers are left?

Done

2.1

Column addition with bigger numbers

Year 5
Add whole numbers with more than 4 digits using the column method

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Column addition with bigger numbersComing soon

In Year 4 you added numbers with up to four digits. Now the numbers have five or six digits, so they are in the tens of thousands and the hundreds of thousands. The column method works in exactly the same way, there are just more columns. Write one number under the other so that digits with the same value are in the same column. Then add one column at a time, starting on the right with the ones. The pale line in the table separates the thousands from the hundreds, to help you read big numbers.

	TTh	Th	H	T	O
	3	4	5	2	6
+	2	1	3	4	2
	5	5	8	6	8

34,526 + 21,342 = 55,868. Each column is added on its own. No column adds up to 10 or more, so there is nothing to carry.

Carrying. Sometimes a column adds up to 10 or more. There is only room for one digit in each column, so we carry. 10 of one column make 1 of the next column on the left: 10 tens make 1 hundred, 10 thousands make 1 ten thousand, and 10 ten thousands make 1 hundred thousand. Write the ones digit of the column total under the line and the carried digit small, under the next column.

10 ten thousands make 1 hundred thousand. The same idea works in every column.

	TTh	Th	H	T	O
	5	6	4	7	8
+	2	7	3	5	5
	8	3	8	3	3
	1		1	1

56,478 + 27,355 = 83,833. The small red digits are the carries. A carried digit is added in with the next column.

Numbers with different lengths. You can add a four-digit number to a five-digit number, like 45,208 + 3,796. Line the digits up on the right, so that ones are under ones. Where a number has no digit, there is nothing to add in that column. Do not line numbers up on the left!

Adding more than two numbers. Put the numbers one under another and add each column from the top to the bottom. When there are three numbers the column total can be as big as 27, so a carry can be 1 or 2.

	TTh	Th	H	T	O
		8	4	5	0
+	7	6	2	9	3
+			5	0	4
	8	5	2	4	7
	1	1	1

8,450 + 76,293 + 504 = 85,247. Three numbers of different lengths, lined up on the right.

The answer can have more digits than either number. If the biggest column adds to 10 or more, the answer gets a new column. For example 674,592 + 538,716 is more than a million, so the answer has seven digits. Always look at the answer at the end and ask: is it a sensible size?

Key words. The column method is a written way to add or subtract. To carry is to move 10 of one column over as 1 of the next column. A place holder is a 0 that keeps a column filled. The sum or total is the answer to an addition.

Key ideas

Write the numbers in columns, with ones under ones, tens under tens, and so on. Line them up on the right.
Add the ones first, then work to the left: tens, hundreds, thousands, ten thousands, hundred thousands.
If a column adds up to 10 or more, write the ones digit and carry the rest. Add the carried digit in the next column.
With three numbers the carry can be 2. Add the carry in as well.
Check with the inverse (opposite) operation: take one of the numbers away from your answer.

Worked example 2.1.1

Use the column method to work out 34,526 + 21,342.

Key words. Column addition is a written method. You write the numbers one under the other, with ones under ones, tens under tens and so on, and add one column at a time. When a column adds up to 10 or more, you carry to the next column. The sum or total is the answer to an addition.

Method. Write the numbers in columns with the same place values lined up. Add the ones first, then the tens, hundreds and so on. If a column makes 10 or more, carry to the next column.

Line up the digits in columns. Ones go under ones, tens under tens, hundreds under hundreds and so on, so every digit is added to digits of the same value.
TTh Th H T O
3 4 5 2 6
+ 2 1 3 4 2
Add the ones. Always start with the ones, on the right.
6 + 2 = 8
8 is less than 10, so write 8 in the ones column. There is nothing to carry.
TTh Th H T O
3 4 5 2 6
+ 2 1 3 4 2
8
Add the tens. Now the next column to the left.
2 + 4 = 6
6 is less than 10, so write 6 in the tens column. There is nothing to carry.
TTh Th H T O
3 4 5 2 6
+ 2 1 3 4 2
6 8
Add the hundreds. Now the next column to the left.
5 + 3 = 8
8 is less than 10, so write 8 in the hundreds column. There is nothing to carry.
TTh Th H T O
3 4 5 2 6
+ 2 1 3 4 2
8 6 8
Add the thousands. Now the next column to the left.
4 + 1 = 5
5 is less than 10, so write 5 in the thousands column. There is nothing to carry.
TTh Th H T O
3 4 5 2 6
+ 2 1 3 4 2
5 8 6 8
Add the ten thousands. Now the next column to the left.
3 + 2 = 5
5 is less than 10, so write 5 in the ten thousands column. There is nothing to carry.
TTh Th H T O
3 4 5 2 6
+ 2 1 3 4 2
5 5 8 6 8
Read the answer. Check that the answer has a sensible number of digits, then write it.
TTh Th H T O
3 4 5 2 6
+ 2 1 3 4 2
5 5 8 6 8

34,526 + 21,342 = 55,868

Check. Use the opposite (inverse) operation: 55,868 − 21,342 = 34,526. ✓
Estimate: 35,000 + 21,000 = 56,000 (rounding to the nearest 1,000). It is close to the exact answer. ✓

Watch out. Start on the right with the ones, not on the left.

Worked example 2.1.2

Use the column method to work out 56,478 + 27,355.

Method. Write the numbers in columns with the same place values lined up. Add the ones first, then the tens, hundreds and so on. If a column makes 10 or more, carry to the next column.

Line up the digits in columns. Ones go under ones, tens under tens, hundreds under hundreds and so on, so every digit is added to digits of the same value.
TTh Th H T O
5 6 4 7 8
+ 2 7 3 5 5
Add the ones. Always start with the ones, on the right.
8 + 5 = 13
13 ones is 1 ten and 3 ones. Write 3 in the ones column and carry 1 to the tens column.
TTh Th H T O
5 6 4 7 8
+ 2 7 3 5 5
3
1
Add the tens. Now the next column to the left. Add the 1 we carried as well.
7 + 5 + 1 = 13 (the red digit is the carry)
13 tens is 1 hundred and 3 tens. Write 3 in the tens column and carry 1 to the hundreds column.
TTh Th H T O
5 6 4 7 8
+ 2 7 3 5 5
3 3
1 1
Add the hundreds. Now the next column to the left. Add the 1 we carried as well.
4 + 3 + 1 = 8 (the red digit is the carry)
8 is less than 10, so write 8 in the hundreds column. There is nothing to carry.
TTh Th H T O
5 6 4 7 8
+ 2 7 3 5 5
8 3 3
1 1
Add the thousands. Now the next column to the left.
6 + 7 = 13
13 thousands is 1 ten thousand and 3 thousands. Write 3 in the thousands column and carry 1 to the ten thousands column.
TTh Th H T O
5 6 4 7 8
+ 2 7 3 5 5
3 8 3 3
1 1 1
Add the ten thousands. Now the next column to the left. Add the 1 we carried as well.
5 + 2 + 1 = 8 (the red digit is the carry)
8 is less than 10, so write 8 in the ten thousands column. There is nothing to carry.
TTh Th H T O
5 6 4 7 8
+ 2 7 3 5 5
8 3 8 3 3
1 1 1
Read the answer. Check that the answer has a sensible number of digits, then write it.
TTh Th H T O
5 6 4 7 8
+ 2 7 3 5 5
8 3 8 3 3
1 1 1

56,478 + 27,355 = 83,833

Check. Use the opposite (inverse) operation: 83,833 − 27,355 = 56,478. ✓
Estimate: 56,000 + 27,000 = 83,000 (rounding to the nearest 1,000). It is close to the exact answer. ✓

Watch out. Do not forget the carried digits. Write each one small under the answer line, and add it in the next column.

Worked example 2.1.3

Use the column method to work out 358,764 + 247,689.

Method. Write the numbers in columns with the same place values lined up. Add the ones first, then the tens, hundreds and so on. If a column makes 10 or more, carry to the next column.

Line up the digits in columns. Ones go under ones, tens under tens, hundreds under hundreds and so on, so every digit is added to digits of the same value.
HTh TTh Th H T O
3 5 8 7 6 4
+ 2 4 7 6 8 9
Add the ones. Always start with the ones, on the right.
4 + 9 = 13
13 ones is 1 ten and 3 ones. Write 3 in the ones column and carry 1 to the tens column.
HTh TTh Th H T O
3 5 8 7 6 4
+ 2 4 7 6 8 9
3
1
Add the tens. Now the next column to the left. Add the 1 we carried as well.
6 + 8 + 1 = 15 (the red digit is the carry)
15 tens is 1 hundred and 5 tens. Write 5 in the tens column and carry 1 to the hundreds column.
HTh TTh Th H T O
3 5 8 7 6 4
+ 2 4 7 6 8 9
5 3
1 1
Add the hundreds. Now the next column to the left. Add the 1 we carried as well.
7 + 6 + 1 = 14 (the red digit is the carry)
14 hundreds is 1 thousand and 4 hundreds. Write 4 in the hundreds column and carry 1 to the thousands column.
HTh TTh Th H T O
3 5 8 7 6 4
+ 2 4 7 6 8 9
4 5 3
1 1 1
Add the thousands. Now the next column to the left. Add the 1 we carried as well.
8 + 7 + 1 = 16 (the red digit is the carry)
16 thousands is 1 ten thousand and 6 thousands. Write 6 in the thousands column and carry 1 to the ten thousands column.
HTh TTh Th H T O
3 5 8 7 6 4
+ 2 4 7 6 8 9
6 4 5 3
1 1 1 1
Add the ten thousands. Now the next column to the left. Add the 1 we carried as well.
5 + 4 + 1 = 10 (the red digit is the carry)
10 ten thousands is 1 hundred thousand and 0 ten thousands. Write 0 in the ten thousands column and carry 1 to the hundred thousands column.
HTh TTh Th H T O
3 5 8 7 6 4
+ 2 4 7 6 8 9
0 6 4 5 3
1 1 1 1 1
Add the hundred thousands. Now the next column to the left. Add the 1 we carried as well.
3 + 2 + 1 = 6 (the red digit is the carry)
6 is less than 10, so write 6 in the hundred thousands column. There is nothing to carry.
HTh TTh Th H T O
3 5 8 7 6 4
+ 2 4 7 6 8 9
6 0 6 4 5 3
1 1 1 1 1
Read the answer. Check that the answer has a sensible number of digits, then write it.
HTh TTh Th H T O
3 5 8 7 6 4
+ 2 4 7 6 8 9
6 0 6 4 5 3
1 1 1 1 1

358,764 + 247,689 = 606,453

Check. Use the opposite (inverse) operation: 606,453 − 247,689 = 358,764. ✓
Estimate: 360,000 + 250,000 = 610,000 (rounding to the nearest 10,000). It is close to the exact answer. ✓

Watch out. Do not forget the carried digits. Write each one small under the answer line, and add it in the next column.

Worked example 2.1.4

Use the column method to work out 45,208 + 3,796. Take care to line the digits up.

Method. Write the numbers in columns with the same place values lined up. Add the ones first, then the tens, hundreds and so on. If a column makes 10 or more, carry to the next column.

Line up the digits in columns. Ones go under ones, tens under tens, hundreds under hundreds and so on, so every digit is added to digits of the same value. The numbers have different lengths, so line them up on the right.
TTh Th H T O
4 5 2 0 8
+ 3 7 9 6
Add the ones. Always start with the ones, on the right.
8 + 6 = 14
14 ones is 1 ten and 4 ones. Write 4 in the ones column and carry 1 to the tens column.
TTh Th H T O
4 5 2 0 8
+ 3 7 9 6
4
1
Add the tens. Now the next column to the left. Add the 1 we carried as well.
0 + 9 + 1 = 10 (the red digit is the carry)
10 tens is 1 hundred and 0 tens. Write 0 in the tens column and carry 1 to the hundreds column.
TTh Th H T O
4 5 2 0 8
+ 3 7 9 6
0 4
1 1
Add the hundreds. Now the next column to the left. Add the 1 we carried as well.
2 + 7 + 1 = 10 (the red digit is the carry)
10 hundreds is 1 thousand and 0 hundreds. Write 0 in the hundreds column and carry 1 to the thousands column.
TTh Th H T O
4 5 2 0 8
+ 3 7 9 6
0 0 4
1 1 1
Add the thousands. Now the next column to the left. Add the 1 we carried as well.
5 + 3 + 1 = 9 (the red digit is the carry)
9 is less than 10, so write 9 in the thousands column. There is nothing to carry.
TTh Th H T O
4 5 2 0 8
+ 3 7 9 6
9 0 0 4
1 1 1
Bring down the ten thousands digit. The other numbers have no digit in this column, so there is nothing to add.
There is only one digit here, so write 4 in the ten thousands column.
TTh Th H T O
4 5 2 0 8
+ 3 7 9 6
4 9 0 0 4
1 1 1
Read the answer. Check that the answer has a sensible number of digits, then write it.
TTh Th H T O
4 5 2 0 8
+ 3 7 9 6
4 9 0 0 4
1 1 1

45,208 + 3,796 = 49,004

Check. Use the opposite (inverse) operation: 49,004 − 3,796 = 45,208. ✓
Estimate: 45,200 + 3,800 = 49,000 (rounding to the nearest 100). It is close to the exact answer. ✓

Watch out. Do not forget the carried digits. Write each one small under the answer line, and add it in the next column.

Worked example 2.1.5

Use the column method to work out 306,040 + 95,960.

Method. Write the numbers in columns with the same place values lined up. Add the ones first, then the tens, hundreds and so on. If a column makes 10 or more, carry to the next column.

Line up the digits in columns. Ones go under ones, tens under tens, hundreds under hundreds and so on, so every digit is added to digits of the same value. The numbers have different lengths, so line them up on the right.
HTh TTh Th H T O
3 0 6 0 4 0
+ 9 5 9 6 0
Add the ones. Always start with the ones, on the right.
0 + 0 = 0
0 is less than 10, so write 0 in the ones column. There is nothing to carry.
Add the tens. Now the next column to the left.
4 + 6 = 10
10 tens is 1 hundred and 0 tens. Write 0 in the tens column and carry 1 to the hundreds column.
Add the hundreds. Now the next column to the left. Add the 1 we carried as well.
0 + 9 + 1 = 10 (the red digit is the carry)
10 hundreds is 1 thousand and 0 hundreds. Write 0 in the hundreds column and carry 1 to the thousands column.
Add the thousands. Now the next column to the left. Add the 1 we carried as well.
6 + 5 + 1 = 12 (the red digit is the carry)
12 thousands is 1 ten thousand and 2 thousands. Write 2 in the thousands column and carry 1 to the ten thousands column.
Add the ten thousands. Now the next column to the left. Add the 1 we carried as well.
0 + 9 + 1 = 10 (the red digit is the carry)
10 ten thousands is 1 hundred thousand and 0 ten thousands. Write 0 in the ten thousands column and carry 1 to the hundred thousands column.
Add the hundred thousands. Now the next column to the left. Add the 1 we carried as well.
3 + 1 = 4 (the red digit is the carry)
4 is less than 10, so write 4 in the hundred thousands column. There is nothing to carry.
Read the answer. Check that the answer has a sensible number of digits, then write it.
HTh TTh Th H T O
3 0 6 0 4 0
+ 9 5 9 6 0
4 0 2 0 0 0
1 1 1 1

306,040 + 95,960 = 402,000

Check. Use the opposite (inverse) operation: 402,000 − 95,960 = 306,040. ✓
Estimate: 306,000 + 96,000 = 402,000 (rounding to the nearest 1,000). It is close to the exact answer. ✓

Watch out. Do not forget the carried digits. Write each one small under the answer line, and add it in the next column.

Worked example 2.1.6

Use the column method to work out 8,450 + 76,293 + 504.

Method. Write the numbers in columns with the same place values lined up. Add the ones first, then the tens, hundreds and so on. If a column makes 10 or more, carry to the next column.

Line up the digits in columns. Ones go under ones, tens under tens, hundreds under hundreds and so on, so every digit is added to digits of the same value. The numbers have different lengths, so line them up on the right.
TTh Th H T O
8 4 5 0
+ 7 6 2 9 3
+ 5 0 4
Add the ones. Always start with the ones, on the right.
0 + 3 + 4 = 7
7 is less than 10, so write 7 in the ones column. There is nothing to carry.
Add the tens. Now the next column to the left.
5 + 9 + 0 = 14
14 tens is 1 hundred and 4 tens. Write 4 in the tens column and carry 1 to the hundreds column.
Add the hundreds. Now the next column to the left. Add the 1 we carried as well.
4 + 2 + 5 + 1 = 12 (the red digit is the carry)
12 hundreds is 1 thousand and 2 hundreds. Write 2 in the hundreds column and carry 1 to the thousands column.
Add the thousands. Now the next column to the left. Add the 1 we carried as well.
8 + 6 + 1 = 15 (the red digit is the carry)
15 thousands is 1 ten thousand and 5 thousands. Write 5 in the thousands column and carry 1 to the ten thousands column.
Add the ten thousands. Now the next column to the left. Add the 1 we carried as well.
7 + 1 = 8 (the red digit is the carry)
8 is less than 10, so write 8 in the ten thousands column. There is nothing to carry.
Read the answer. Check that the answer has a sensible number of digits, then write it.
TTh Th H T O
8 4 5 0
+ 7 6 2 9 3
+ 5 0 4
8 5 2 4 7
1 1 1

8,450 + 76,293 + 504 = 85,247

Check. Use the opposite (inverse) operation: 85,247 − 76,293 − 504 = 8,450. ✓
Estimate: 8,450 + 76,290 + 500 = 85,240 (rounding to the nearest 10). It is close to the exact answer. ✓

Watch out. Do not forget the carried digits. Write each one small under the answer line, and add it in the next column.

Worked example 2.1.7

Use the column method to work out 674,592 + 538,716.

Method. Write the numbers in columns with the same place values lined up. Add the ones first, then the tens, hundreds and so on. If a column makes 10 or more, carry to the next column.

Line up the digits in columns. Ones go under ones, tens under tens, hundreds under hundreds and so on, so every digit is added to digits of the same value.
M HTh TTh Th H T O
6 7 4 5 9 2
+ 5 3 8 7 1 6
Add the ones. Always start with the ones, on the right.
2 + 6 = 8
8 is less than 10, so write 8 in the ones column. There is nothing to carry.
Add the tens. Now the next column to the left.
9 + 1 = 10
10 tens is 1 hundred and 0 tens. Write 0 in the tens column and carry 1 to the hundreds column.
Add the hundreds. Now the next column to the left. Add the 1 we carried as well.
5 + 7 + 1 = 13 (the red digit is the carry)
13 hundreds is 1 thousand and 3 hundreds. Write 3 in the hundreds column and carry 1 to the thousands column.
Add the thousands. Now the next column to the left. Add the 1 we carried as well.
4 + 8 + 1 = 13 (the red digit is the carry)
13 thousands is 1 ten thousand and 3 thousands. Write 3 in the thousands column and carry 1 to the ten thousands column.
Add the ten thousands. Now the next column to the left. Add the 1 we carried as well.
7 + 3 + 1 = 11 (the red digit is the carry)
11 ten thousands is 1 hundred thousand and 1 ten thousand. Write 1 in the ten thousands column and carry 1 to the hundred thousands column.
Add the hundred thousands. Now the next column to the left. Add the 1 we carried as well.
6 + 5 + 1 = 12 (the red digit is the carry)
12 hundred thousands is 1 million and 2 hundred thousands. Write 2 in the hundred thousands column and carry 1 to the millions column.
Write the carried digit. There are no digits left in the millions column, but we carried 1 into it.
There is only one digit here, so write 1 in the millions column.
Read the answer. Check that the answer has a sensible number of digits, then write it.
M HTh TTh Th H T O
6 7 4 5 9 2
+ 5 3 8 7 1 6
1 2 1 3 3 0 8
1 1 1 1 1

674,592 + 538,716 = 1,213,308

Check. Use the opposite (inverse) operation: 1,213,308 − 538,716 = 674,592. ✓
Estimate: 670,000 + 540,000 = 1,210,000 (rounding to the nearest 10,000). It is close to the exact answer. ✓

Watch out. Do not forget the carried digits. Write each one small under the answer line, and add it in the next column.

Worked example 2.1.8

Over two seasons, 248,350 fans watch Rovers at home in the first season and 135,670 fans watch in the second season. How many fans watch altogether?

Method. Draw a bar model to see what is happening. Decide whether to add or subtract, then use a written method.

Draw a bar model. A picture shows what we know and what we need to find.
Choose the operation. We know both parts and want the whole, so we add.
248,350 + 135,670
Work it out. Use the column method because the numbers are big.
248,350 + 135,670 = 384,020
HTh TTh Th H T O
2 4 8 3 5 0
+ 1 3 5 6 7 0
3 8 4 0 2 0
1 1 1
Write the answer.
384,020 fans

Check. Use the opposite operation: 384,020 − 135,670 = 248,350. ✓

Watch out. Read the question again at the end. Does your answer make sense, and does it answer what was asked?

Watch out

Do not forget the carried digit! It is easy to write the carry and then not add it in the next column. Also line the numbers up on the right: in 45,208 + 3,796 the 6 of 3,796 goes under the 8 (ones under ones), not under the 4.

2.2

Column subtraction with bigger numbers

Year 5
Subtract whole numbers with more than 4 digits using the column method

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Column subtraction with bigger numbersComing soon

In column subtraction the bigger number goes on top. It is the number you start with. You take the bottom number away from it, one column at a time, starting with the ones. Words that mean subtract are take away, minus, less than, fewer than and difference.

	TTh	Th	H	T	O
	7	6	8	4	5
−	3	2	6	1	3
	4	4	2	3	2

76,845 − 32,613 = 44,232. Every top digit is big enough, so nothing needs to be exchanged.

Exchanging. Sometimes the top digit is smaller than the bottom digit, like 2 − 7. You cannot take 7 away from 2. So you exchange: swap 1 from the next column on the left for 10 in this column. The digit on the left goes down by 1 and this digit goes up by 10. This works in every column. 1 hundred thousand is 10 ten thousands, and 1 ten thousand is 10 thousands.

Exchange 1 hundred thousand for 10 ten thousands. The value of the number does not change.

	TTh	Th	H	T	O
	56	133	34	145	122
−	2	8	1	7	6
	3	5	2	7	6

63,452 − 28,176 = 35,276. The small red digits are the new digits after an exchange. The old digits are crossed out.

Zeros are tricky. You cannot exchange from a 0, because there is nothing to swap. Go left to the first column that is not 0, and pass the exchange along one column at a time. In 500,000 − 237,894 the ones need help, but every column on the left is 0 until the hundred thousands. So 1 hundred thousand becomes 10 ten thousands, then 1 ten thousand becomes 10 thousands, and so on. Each 0 in the middle ends up as a 9.

	HTh	TTh	Th	H	T	O
	45	90	90	90	90	100
−	2	3	7	8	9	4
	2	6	2	1	0	6

500,000 − 237,894 = 262,106. The 5 becomes 4 and all the zeros in the middle become 9s. The last 0 becomes 10.

Different lengths. You can take a four-digit number from a six-digit number, like 462,310 − 5,874. Line the digits up on the right. Where the bottom number has no digit, there is nothing to take away. And 1,000,000 is a seven-digit number: 1,000,000 − 468,325 works exactly like the 500,000 example, with six zeros to pass along.

Key words. To exchange is to swap 1 of a bigger unit for 10 of the next smaller unit (some people call it borrowing). The difference between two numbers is how far apart they are. It is the answer to a subtraction.

Key ideas

The bigger number goes on top. Line the digits up on the right.
Take away the ones first, then work left: tens, hundreds, thousands, ten thousands and so on.
If the top digit is smaller than the bottom digit, exchange: take 1 from the column on the left and add 10 to this column.
You cannot exchange from a 0. Go left to the first column that is not 0. Each 0 you pass becomes 9.
Check with addition: the answer plus the number you took away gives the number you started with.

Worked example 2.2.1

Use the column method to work out 76,845 − 32,613.

Method. Write the bigger number on top. Take away the ones first, then the tens, hundreds and so on. If a top digit is too small, exchange 1 from the next column.

Line up the digits in columns. The bigger number goes on top. Ones go under ones, tens under tens, hundreds under hundreds and so on.
TTh Th H T O
7 6 8 4 5
− 3 2 6 1 3
Subtract the ones. Always start with the ones, on the right.
5 − 3 = 2
Subtract the tens. Now the next column to the left.
4 − 1 = 3
Subtract the hundreds. Now the next column to the left.
8 − 6 = 2
Subtract the thousands. Now the next column to the left.
6 − 2 = 4
Subtract the ten thousands. Now the next column to the left.
7 − 3 = 4
Read the answer. Check that the answer has a sensible number of digits, then write it.
TTh Th H T O
7 6 8 4 5
− 3 2 6 1 3
4 4 2 3 2

76,845 − 32,613 = 44,232

Check. Use the opposite (inverse) operation: 44,232 + 32,613 = 76,845. ✓
Estimate: 77,000 − 33,000 = 44,000 (rounding to the nearest 1,000). It is close to the exact answer. ✓

Watch out. The bigger number goes on top and the digits are taken away column by column, starting with the ones.

Worked example 2.2.2

Use the column method to work out 63,452 − 28,176.

Method. Write the bigger number on top. Take away the ones first, then the tens, hundreds and so on. If a top digit is too small, exchange 1 from the next column.

Line up the digits in columns. The bigger number goes on top. Ones go under ones, tens under tens, hundreds under hundreds and so on.
TTh Th H T O
6 3 4 5 2
− 2 8 1 7 6
Exchange, then subtract the ones. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
2 ones is not enough to take away 6. We need to exchange.
Exchange 1 ten for 10 ones: 5 tens become 4 tens, and 2 ones become 12 ones.
12 − 6 = 6
TTh Th H T O
6 3 4 45 122
− 2 8 1 7 6
6
Exchange, then subtract the tens. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
4 tens is not enough to take away 7. We need to exchange.
(There were 5 tens to start with, but we gave one away in an earlier exchange.)
Exchange 1 hundred for 10 tens: 4 hundreds become 3 hundreds, and 4 tens become 14 tens.
14 − 7 = 7
TTh Th H T O
6 3 34 145 122
− 2 8 1 7 6
7 6
Subtract the hundreds. Now the next column to the left.
(This digit is 3 now, not 4, because we exchanged from this column.)
3 − 1 = 2
TTh Th H T O
6 3 34 145 122
− 2 8 1 7 6
2 7 6
Exchange, then subtract the thousands. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
3 thousands is not enough to take away 8. We need to exchange.
Exchange 1 ten thousand for 10 thousands: 6 ten thousands become 5 ten thousands, and 3 thousands become 13 thousands.
13 − 8 = 5
TTh Th H T O
56 133 34 145 122
− 2 8 1 7 6
5 2 7 6
Subtract the ten thousands. Now the next column to the left.
(This digit is 5 now, not 6, because we exchanged from this column.)
5 − 2 = 3
TTh Th H T O
56 133 34 145 122
− 2 8 1 7 6
3 5 2 7 6
Read the answer. Check that the answer has a sensible number of digits, then write it.
TTh Th H T O
56 133 34 145 122
− 2 8 1 7 6
3 5 2 7 6

63,452 − 28,176 = 35,276

Check. Use the opposite (inverse) operation: 35,276 + 28,176 = 63,452. ✓
Estimate: 63,000 − 28,000 = 35,000 (rounding to the nearest 1,000). It is close to the exact answer. ✓

Watch out. The bottom digit is always taken from the top digit. Do not turn it round: 3 − 7 is not 4. Exchange instead.

Worked example 2.2.3

Use the column method to work out 704,826 − 358,947.

Method. Write the bigger number on top. Take away the ones first, then the tens, hundreds and so on. If a top digit is too small, exchange 1 from the next column.

Line up the digits in columns. The bigger number goes on top. Ones go under ones, tens under tens, hundreds under hundreds and so on.
HTh TTh Th H T O
7 0 4 8 2 6
− 3 5 8 9 4 7
Exchange, then subtract the ones. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
6 ones is not enough to take away 7. We need to exchange.
Exchange 1 ten for 10 ones: 2 tens become 1 ten, and 6 ones become 16 ones.
16 − 7 = 9
Exchange, then subtract the tens. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
1 ten is not enough to take away 4. We need to exchange.
(There were 2 tens to start with, but we gave one away in an earlier exchange.)
Exchange 1 hundred for 10 tens: 8 hundreds become 7 hundreds, and 1 ten become 11 tens.
11 − 4 = 7
Exchange, then subtract the hundreds. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
7 hundreds is not enough to take away 9. We need to exchange.
(There were 8 hundreds to start with, but we gave one away in an earlier exchange.)
Exchange 1 thousand for 10 hundreds: 4 thousands become 3 thousands, and 7 hundreds become 17 hundreds.
17 − 9 = 8
Exchange, then subtract the thousands. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
3 thousands is not enough to take away 8. We need to exchange.
(There were 4 thousands to start with, but we gave one away in an earlier exchange.)
The ten thousands has 0, so there is nothing to exchange from there. We start further to the left, at the hundred thousands.
Exchange 1 hundred thousand for 10 ten thousands: 7 hundred thousands become 6 hundred thousands, and 0 ten thousands become 10 ten thousands.
Exchange 1 ten thousand for 10 thousands: 10 ten thousands become 9 ten thousands, and 3 thousands become 13 thousands.
13 − 8 = 5
Subtract the ten thousands. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 5 = 4
Subtract the hundred thousands. Now the next column to the left.
(This digit is 6 now, not 7, because we exchanged from this column.)
6 − 3 = 3
Read the answer. Check that the answer has a sensible number of digits, then write it.
HTh TTh Th H T O
67 90 134 178 112 166
− 3 5 8 9 4 7
3 4 5 8 7 9

704,826 − 358,947 = 345,879

Check. Use the opposite (inverse) operation: 345,879 + 358,947 = 704,826. ✓
Estimate: 700,000 − 360,000 = 340,000 (rounding to the nearest 10,000). It is close to the exact answer. ✓

Watch out. You cannot exchange from a 0. Look left for the first column that is not 0, then pass the exchange along one column at a time. Each 0 in the middle becomes 9.

Worked example 2.2.4

Use the column method to work out 80,003 − 24,567.

Method. Write the bigger number on top. Take away the ones first, then the tens, hundreds and so on. If a top digit is too small, exchange 1 from the next column.

Line up the digits in columns. The bigger number goes on top. Ones go under ones, tens under tens, hundreds under hundreds and so on.
TTh Th H T O
8 0 0 0 3
− 2 4 5 6 7
Exchange, then subtract the ones. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
3 ones is not enough to take away 7. We need to exchange.
The tens, hundreds and thousands have 0, so there is nothing to exchange from there. We start further to the left, at the ten thousands.
Exchange 1 ten thousand for 10 thousands: 8 ten thousands become 7 ten thousands, and 0 thousands become 10 thousands.
Exchange 1 thousand for 10 hundreds: 10 thousands become 9 thousands, and 0 hundreds become 10 hundreds.
Exchange 1 hundred for 10 tens: 10 hundreds become 9 hundreds, and 0 tens become 10 tens.
Exchange 1 ten for 10 ones: 10 tens become 9 tens, and 3 ones become 13 ones.
13 − 7 = 6
Subtract the tens. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 6 = 3
Subtract the hundreds. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 5 = 4
Subtract the thousands. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 4 = 5
Subtract the ten thousands. Now the next column to the left.
(This digit is 7 now, not 8, because we exchanged from this column.)
7 − 2 = 5
Read the answer. Check that the answer has a sensible number of digits, then write it.
TTh Th H T O
78 90 90 90 133
− 2 4 5 6 7
5 5 4 3 6

80,003 − 24,567 = 55,436

Check. Use the opposite (inverse) operation: 55,436 + 24,567 = 80,003. ✓
Estimate: 80,000 − 25,000 = 55,000 (rounding to the nearest 1,000). It is close to the exact answer. ✓

Watch out. You cannot exchange from a 0. Look left for the first column that is not 0, then pass the exchange along one column at a time. Each 0 in the middle becomes 9.

Worked example 2.2.5

Use the column method to work out 500,000 − 237,894.

Method. Write the bigger number on top. Take away the ones first, then the tens, hundreds and so on. If a top digit is too small, exchange 1 from the next column.

Line up the digits in columns. The bigger number goes on top. Ones go under ones, tens under tens, hundreds under hundreds and so on.
HTh TTh Th H T O
5 0 0 0 0 0
− 2 3 7 8 9 4
Exchange, then subtract the ones. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
0 ones is not enough to take away 4. We need to exchange.
The tens, hundreds, thousands and ten thousands have 0, so there is nothing to exchange from there. We start further to the left, at the hundred thousands.
Exchange 1 hundred thousand for 10 ten thousands: 5 hundred thousands become 4 hundred thousands, and 0 ten thousands become 10 ten thousands.
Exchange 1 ten thousand for 10 thousands: 10 ten thousands become 9 ten thousands, and 0 thousands become 10 thousands.
Exchange 1 thousand for 10 hundreds: 10 thousands become 9 thousands, and 0 hundreds become 10 hundreds.
Exchange 1 hundred for 10 tens: 10 hundreds become 9 hundreds, and 0 tens become 10 tens.
Exchange 1 ten for 10 ones: 10 tens become 9 tens, and 0 ones become 10 ones.
10 − 4 = 6
HTh TTh Th H T O
45 90 90 90 90 100
− 2 3 7 8 9 4
6
Subtract the tens. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 9 = 0
HTh TTh Th H T O
45 90 90 90 90 100
− 2 3 7 8 9 4
0 6
Subtract the hundreds. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 8 = 1
HTh TTh Th H T O
45 90 90 90 90 100
− 2 3 7 8 9 4
1 0 6
Subtract the thousands. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 7 = 2
HTh TTh Th H T O
45 90 90 90 90 100
− 2 3 7 8 9 4
2 1 0 6
Subtract the ten thousands. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 3 = 6
HTh TTh Th H T O
45 90 90 90 90 100
− 2 3 7 8 9 4
6 2 1 0 6
Subtract the hundred thousands. Now the next column to the left.
(This digit is 4 now, not 5, because we exchanged from this column.)
4 − 2 = 2
HTh TTh Th H T O
45 90 90 90 90 100
− 2 3 7 8 9 4
2 6 2 1 0 6
Read the answer. Check that the answer has a sensible number of digits, then write it.
HTh TTh Th H T O
45 90 90 90 90 100
− 2 3 7 8 9 4
2 6 2 1 0 6

500,000 − 237,894 = 262,106

Check. Use the opposite (inverse) operation: 262,106 + 237,894 = 500,000. ✓
Estimate: 500,000 − 240,000 = 260,000 (rounding to the nearest 10,000). It is close to the exact answer. ✓

Watch out. You cannot exchange from a 0. Look left for the first column that is not 0, then pass the exchange along one column at a time. Each 0 in the middle becomes 9.

Worked example 2.2.6

Use the column method to work out 462,310 − 5,874. Take care to line the digits up.

Method. Write the bigger number on top. Take away the ones first, then the tens, hundreds and so on. If a top digit is too small, exchange 1 from the next column.

Line up the digits in columns. The bigger number goes on top. Ones go under ones, tens under tens, hundreds under hundreds and so on. The numbers have different lengths, so line them up on the right.
HTh TTh Th H T O
4 6 2 3 1 0
− 5 8 7 4
Exchange, then subtract the ones. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
0 ones is not enough to take away 4. We need to exchange.
Exchange 1 ten for 10 ones: 1 ten become 0 tens, and 0 ones become 10 ones.
10 − 4 = 6
Exchange, then subtract the tens. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
0 tens is not enough to take away 7. We need to exchange.
(There were 1 tens to start with, but we gave one away in an earlier exchange.)
Exchange 1 hundred for 10 tens: 3 hundreds become 2 hundreds, and 0 tens become 10 tens.
10 − 7 = 3
Exchange, then subtract the hundreds. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
2 hundreds is not enough to take away 8. We need to exchange.
(There were 3 hundreds to start with, but we gave one away in an earlier exchange.)
Exchange 1 thousand for 10 hundreds: 2 thousands become 1 thousand, and 2 hundreds become 12 hundreds.
12 − 8 = 4
Exchange, then subtract the thousands. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
1 thousand is not enough to take away 5. We need to exchange.
(There were 2 thousands to start with, but we gave one away in an earlier exchange.)
Exchange 1 ten thousand for 10 thousands: 6 ten thousands become 5 ten thousands, and 1 thousand become 11 thousands.
11 − 5 = 6
Bring down the ten thousands digit. The bottom number has no digit in this column, so there is nothing to take away.
(This digit is 5 now, not 6, because we exchanged from this column.)
Nothing to take away in this column, so the digit stays: 5.
Bring down the hundred thousands digit. The bottom number has no digit in this column, so there is nothing to take away.
Nothing to take away in this column, so the digit stays: 4.
Read the answer. Check that the answer has a sensible number of digits, then write it.
HTh TTh Th H T O
4 56 112 123 101 100
− 5 8 7 4
4 5 6 4 3 6

462,310 − 5,874 = 456,436

Check. Use the opposite (inverse) operation: 456,436 + 5,874 = 462,310. ✓
Estimate: 462,300 − 5,900 = 456,400 (rounding to the nearest 100). It is close to the exact answer. ✓

Watch out. The bottom digit is always taken from the top digit. Do not turn it round: 3 − 7 is not 4. Exchange instead.

Worked example 2.2.7

Use the column method to work out 1,000,000 − 468,325.

Method. Write the bigger number on top. Take away the ones first, then the tens, hundreds and so on. If a top digit is too small, exchange 1 from the next column.

Line up the digits in columns. The bigger number goes on top. Ones go under ones, tens under tens, hundreds under hundreds and so on. The numbers have different lengths, so line them up on the right.
M HTh TTh Th H T O
1 0 0 0 0 0 0
− 4 6 8 3 2 5
Exchange, then subtract the ones. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
0 ones is not enough to take away 5. We need to exchange.
The tens, hundreds, thousands, ten thousands and hundred thousands have 0, so there is nothing to exchange from there. We start further to the left, at the millions.
Exchange 1 million for 10 hundred thousands: 1 million become 0 millions, and 0 hundred thousands become 10 hundred thousands.
Exchange 1 hundred thousand for 10 ten thousands: 10 hundred thousands become 9 hundred thousands, and 0 ten thousands become 10 ten thousands.
Exchange 1 ten thousand for 10 thousands: 10 ten thousands become 9 ten thousands, and 0 thousands become 10 thousands.
Exchange 1 thousand for 10 hundreds: 10 thousands become 9 thousands, and 0 hundreds become 10 hundreds.
Exchange 1 hundred for 10 tens: 10 hundreds become 9 hundreds, and 0 tens become 10 tens.
Exchange 1 ten for 10 ones: 10 tens become 9 tens, and 0 ones become 10 ones.
10 − 5 = 5
Subtract the tens. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 2 = 7
Subtract the hundreds. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 3 = 6
Subtract the thousands. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 8 = 1
Subtract the ten thousands. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 6 = 3
Subtract the hundred thousands. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 4 = 5
Bring down the millions digit. The bottom number has no digit in this column, so there is nothing to take away.
(This digit is 0 now, not 1, because we exchanged from this column.)
Nothing to take away in this column, so the digit stays: 0.
We do not write a 0 at the front of an answer, so leave this space empty.
Read the answer. Check that the answer has a sensible number of digits, then write it.
M HTh TTh Th H T O
01 90 90 90 90 90 100
− 4 6 8 3 2 5
5 3 1 6 7 5

1,000,000 − 468,325 = 531,675

Check. Use the opposite (inverse) operation: 531,675 + 468,325 = 1,000,000. ✓
Estimate: 1,000,000 − 470,000 = 530,000 (rounding to the nearest 10,000). It is close to the exact answer. ✓

Watch out. You cannot exchange from a 0. Look left for the first column that is not 0, then pass the exchange along one column at a time. Each 0 in the middle becomes 9.

Worked example 2.2.8

Find the difference between 53,206 and 98,114.

Method. Write the bigger number on top. Take away the ones first, then the tens, hundreds and so on. If a top digit is too small, exchange 1 from the next column.

Line up the digits in columns. The bigger number goes on top. Ones go under ones, tens under tens, hundreds under hundreds and so on.
TTh Th H T O
9 8 1 1 4
− 5 3 2 0 6
Exchange, then subtract the ones. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
4 ones is not enough to take away 6. We need to exchange.
Exchange 1 ten for 10 ones: 1 ten become 0 tens, and 4 ones become 14 ones.
14 − 6 = 8
Subtract the tens. Now the next column to the left.
(This digit is 0 now, not 1, because we exchanged from this column.)
0 − 0 = 0
Exchange, then subtract the hundreds. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
1 hundred is not enough to take away 2. We need to exchange.
Exchange 1 thousand for 10 hundreds: 8 thousands become 7 thousands, and 1 hundred become 11 hundreds.
11 − 2 = 9
Subtract the thousands. Now the next column to the left.
(This digit is 7 now, not 8, because we exchanged from this column.)
7 − 3 = 4
Subtract the ten thousands. Now the next column to the left.
9 − 5 = 4
Read the answer. Check that the answer has a sensible number of digits, then write it.
TTh Th H T O
9 78 111 01 144
− 5 3 2 0 6
4 4 9 0 8

98,114 − 53,206 = 44,908

Check. Use the opposite (inverse) operation: 44,908 + 53,206 = 98,114. ✓
Estimate: 98,000 − 53,000 = 45,000 (rounding to the nearest 1,000). It is close to the exact answer. ✓

Watch out. The bottom digit is always taken from the top digit. Do not turn it round: 3 − 7 is not 4. Exchange instead.

Watch out

Never turn the digits round! In 63,452 − 28,176 it is tempting to write 8 − 2 = 6 in the ones column because 2 − 6 is hard. That is wrong. The bottom digit is always taken away from the top digit, so exchange instead. And remember: you cannot exchange from a 0.

2.3

Adding and subtracting in your head

Year 5
Add and subtract numbers mentally with increasingly large numbers

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Adding and subtracting in your headComing soon

A written method always works, but it is not always the quickest. If the numbers are friendly, you can add and subtract in your head. Look at the numbers first, then pick a good strategy. Big numbers are not harder to use in your head: 340,000 + 270,000 is just 34 + 27 with some zeros.

1. Partitioning. Split (partition) a number into its parts and add or take away one part at a time, biggest first. For 36,400 + 25,800: 36,400 + 20,000 = 56,400, then + 5,000 = 61,400, then + 800 = 62,200. To subtract, keep the first number whole and take away the parts of the second number.

2. Bridging. Jump to the next round number (10, 100 or 1,000), then add or subtract what is left. For 5,970 + 2,400, add 30 to reach 6,000, then add the other 2,370.

Bridging through 1,000: 5,970 + 30 = 6,000, and 6,000 + 2,370 = 8,370.

3. Compensating. If a number is just below or above a round number (like 99, 999, 9,999 or 1,999), use the round number and then fix the answer. 36,450 + 9,999: add 10,000 (1 too many) to get 46,450, then take 1 away to get 46,449. To take away 1,999, take away 2,000 and put 1 back.

4. Using number facts. Number bonds to 100 give you bonds to 1,000, 10,000 and 100,000 for free. Because 75 + 25 = 100, we know that 7,500 + 2,500 = 10,000.

Number fact	Bigger fact	Even bigger	Biggest
75 + 25 = 100	750 + 250 = 1,000	7,500 + 2,500 = 10,000	75,000 + 25,000 = 100,000

One number bond to 100 gives a whole family of facts. Each time, both numbers get 10 times bigger.

5. Multiples of 1,000 or 10,000. Count how many thousands (or ten thousands) there are, do the small calculation, and change back. 800,000 − 350,000 is 80 ten thousands − 35 ten thousands = 45 ten thousands = 450,000.

6. Near doubles. If two numbers are close, double the smaller and add the difference. 35,000 + 36,000: double 35,000 is 70,000, plus 1,000 is 71,000.

7. Counting up. To find the difference between two numbers that are close, start at the smaller number and count up to the bigger. For 100,000 − 96,250, count from 96,250 up to 97,000 (750), then on to 100,000 (3,000). The difference is 750 + 3,000 = 3,750.

Counting up from 96,250 to 100,000: 750 + 3,000 = 3,750.

Key words. To partition is to split a number into parts. To bridge is to jump to a round number. To compensate is to make up for a change you made. A number bond is a pair of numbers that make a total. A double is a number added to itself.

Key ideas

Partition: add or take away the parts of the second number one at a time, biggest first.
Bridge: jump to the next 10, 100 or 1,000, then do the rest.
Compensate: use a round number, then fix the answer (add 9,999 = add 10,000, take away 1).
Use number facts: 75 + 25 = 100, so 7,500 + 2,500 = 10,000.
Multiples of 1,000 or 10,000: count the units, calculate, then change back.
Near doubles: double the smaller number and add the difference.
Count up to find the difference when the numbers are close.

Worked example 2.3.1

Work out 36,400 + 25,800 in your head by partitioning.

Key words. To partition a number is to split it into parts, one part for each digit (for example 3,450 = 3,000 + 400 + 50).

Method. Keep the first number whole. Split (partition) the second number into its parts, then add the parts one at a time, biggest first.

Partition the second number. Only the second number is split up.
25,800 = 20,000 + 5,000 + 800
Add the parts one at a time. Start with the biggest part. Each answer is the start of the next step.
36,400 + 20,000 = 56,400
56,400 + 5,000 = 61,400
61,400 + 800 = 62,200
Write the answer. The last number is the answer.
36,400 + 25,800 = 62,200

Check. Use the opposite operation: 62,200 − 25,800 = 36,400. ✓

Watch out. Do not miss a part. Every digit of the second number that is not 0 gives one part to add.

Worked example 2.3.2

Work out 45,200 + 32,700 by splitting both numbers into parts.

Key words. To partition a number is to split it into parts, one part for each digit.

Method. Split (partition) both numbers into thousands, hundreds, tens and ones. Add the parts that match, then add the totals.

Partition both numbers. Write each number as the value of its digits.
45,200 = 40,000 + 5,000 + 200
32,700 = 30,000 + 2,000 + 700
Add the matching parts. Thousands with thousands, hundreds with hundreds, and so on.
ten thousands: 40,000 + 30,000 = 70,000
thousands: 5,000 + 2,000 = 7,000
hundreds: 200 + 700 = 900
Add the totals together. Start with the biggest and keep a running total.
70,000 + 7,000 = 77,000
77,000 + 900 = 77,900
Write the answer.
45,200 + 32,700 = 77,900

Check. Use the opposite operation: 77,900 − 32,700 = 45,200. ✓

Watch out. Do not forget a part. Every digit that is not 0 must be added.

Worked example 2.3.3

Work out 8,764 − 3,425 by partitioning the second number.

Key words. To partition a number is to split it into parts, one part for each digit.

Method. Keep the first number whole. Split (partition) the number you are taking away, then take the parts away one at a time, biggest first.

Partition the number you take away. Only the second number is split up.
3,425 = 3,000 + 400 + 20 + 5
Take the parts away one at a time. Start with the biggest part. Use the new number as the start of the next step.
8,764 − 3,000 = 5,764
5,764 − 400 = 5,364
5,364 − 20 = 5,344
5,344 − 5 = 5,339
Write the answer. The last number is the answer.
8,764 − 3,425 = 5,339

Check. Use the opposite operation: 5,339 + 3,425 = 8,764. ✓

Watch out. Take the parts away from the first number, one after another. Do not take them away from each other.

Worked example 2.3.4

Work out 5,970 + 2,400 by bridging through 1,000.

Key words. To bridge through 1,000 is to use a jump that lands exactly on a multiple of 1,000, then jump again.

Method. Bridge through 1,000. Add enough to reach the next 1,000, then add what is left.

How far is 5,970 from the next 1,000? 6,000 is the next multiple of 1,000.
6,000 − 5,970 = 30
Split 2,400 into 30 and 2,370. The first part takes us exactly to 6,000.
2,400 = 30 + 2,370
Add the first part to reach 6,000. This is the bridge.
5,970 + 30 = 6,000
Add the second part. This is easy because we are on a round number.
6,000 + 2,370 = 8,370
Show it on a number line.

Check. Use the opposite operation: 8,370 − 2,400 = 5,970. ✓

Watch out. Make sure the two parts add back to 2,400: 30 + 2,370 = 2,400.

Worked example 2.3.5

Work out 7,300 − 2,450 by bridging through 1,000.

Method. Bridge through 1,000. Take away enough to land on the 1,000 below, then take away what is left.

How far is 7,300 above the 1,000 below it? 7,000 is the multiple of 1,000 just below 7,300.
7,300 − 7,000 = 300
Split 2,450 into 300 and 2,150. The first part takes us exactly to 7,000.
2,450 = 300 + 2,150
Take away the first part to reach 7,000. This is the bridge.
7,300 − 300 = 7,000
Take away the second part. This is easy because we start from a round number.
7,000 − 2,150 = 4,850
Show it on a number line. We count back from the right.

Check. Use the opposite operation: 4,850 + 2,450 = 7,300. ✓

Watch out. Make sure the two parts add back to 2,450: 300 + 2,150 = 2,450.

Worked example 2.3.6

Work out 36,450 + 9,999 by compensating.

Key words. To compensate is to make up for a change. You use an easier number, then correct the answer.

Method. Compensate: add the nearby round number 10,000 instead, then fix the answer.

Spot a nearby round number. 10,000 is much easier to work with than 9,999.
9,999 is 1 less than 10,000: 9,999 = 10,000 − 1
Add 10,000 instead of 9,999. Round numbers are easy.
36,450 + 10,000 = 46,450
Fix the answer. We added 1 too many, so we must take 1 away.
46,450 − 1 = 46,449

Check. Use the opposite operation: 46,449 − 9,999 = 36,450. ✓

Watch out. Do not forget the last step. You changed the number you add or take away, so the answer needs fixing.

Worked example 2.3.7

Work out 54,300 − 1,999 by compensating.

Key words. To compensate is to make up for a change. You use an easier number, then correct the answer.

Method. Compensate: take away the nearby round number 2,000 instead, then fix the answer.

Spot a nearby round number. 2,000 is much easier to work with than 1,999.
1,999 is 1 less than 2,000: 1,999 = 2,000 − 1
Take away 2,000 instead of 1,999. Round numbers are easy.
54,300 − 2,000 = 52,300
Fix the answer. We took away 1 too many, so we must put 1 back.
52,300 + 1 = 52,301

Check. Use the opposite operation: 52,301 + 1,999 = 54,300. ✓

Watch out. Do not forget the last step. You changed the number you add or take away, so the answer needs fixing.

Worked example 2.3.8

Work out 47,500 + 2,500 using a number fact.

Key words. A number bond is a pair of numbers that add up to a given total, such as 36 + 64 = 100.

Method. Look for a number fact you know. A pair of numbers that make 100 gives a pair that make 1,000, 10,000 or 100,000.

Split 47,500 into a round part and the rest. The rest, 7,500, is the part that goes with 2,500.
47,500 = 40,000 + 7,500
Spot the number bond. Think of 7,500 and 2,500 as 75 and 25 lots of 100. You know that 75 + 25 = 100.
75 + 25 = 100, so 7,500 + 2,500 = 10,000
Put the round numbers together. 10,000 is easy to add to 40,000.
40,000 + 10,000 = 50,000

Check. Use the opposite operation: 50,000 − 2,500 = 47,500. ✓

Watch out. The bond must make 100 (then 1,000, 10,000 and so on). Check: 75 + 25 = 100.

Worked example 2.3.9

Work out 10,000 − 3,600 using a number fact.

Method. Use a number bond to 100. Count how many lots of the small unit there are, then use the fact you know.

Find the unit. 10,000 is 100 lots of 100.
10,000 = 100 × 100
Count the lots in 3,600. 3,600 is 36 lots of 100.
3,600 = 36 × 100
Use the bond to 100. What goes with 36 to make 100?
36 + 64 = 100, so 100 − 36 = 64
Change back to the real numbers. 64 lots of 100.
64 × 100 = 6,400

Check. Use addition: 6,400 + 3,600 = 10,000. ✓

Watch out. Watch the zeros. 100 lots of 100 is 10,000, but 100 lots of 1,000 is 100,000.

Worked example 2.3.10

Work out 340,000 + 270,000 by counting ten thousands.

Method. Both numbers are multiples of 10,000. Count how many ten thousands each one is, do the small calculation, then change back.

Count the ten thousands. 340,000 is 34 ten thousands. 270,000 is 27 ten thousands.
340,000 = 34 × 10,000
270,000 = 27 × 10,000
Do the small calculation. It is just like adding or subtracting 34 and 27.
34 + 27 = 61
(34 + 20 = 54, 54 + 7 = 61)
Change back to 10,000s. 61 ten thousands is 61 × 10,000.
61 × 10,000 = 610,000

Check. Use the opposite operation: 610,000 − 270,000 = 340,000. ✓

Watch out. Count the zeros carefully. 10,000 has 4 zeros, so the answer needs them too.

Worked example 2.3.11

Work out 800,000 − 350,000 by counting ten thousands.

Method. Both numbers are multiples of 10,000. Count how many ten thousands each one is, do the small calculation, then change back.

Count the ten thousands. 800,000 is 80 ten thousands. 350,000 is 35 ten thousands.
800,000 = 80 × 10,000
350,000 = 35 × 10,000
Do the small calculation. It is just like adding or subtracting 80 and 35.
80 − 35 = 45
(80 − 30 = 50, 50 − 5 = 45)
Change back to 10,000s. 45 ten thousands is 45 × 10,000.
45 × 10,000 = 450,000

Check. Use the opposite operation: 450,000 + 350,000 = 800,000. ✓

Watch out. Count the zeros carefully. 10,000 has 4 zeros, so the answer needs them too.

Worked example 2.3.12

Work out 35,000 + 36,000 using a near double.

Key words. A double is a number added to itself. A near double is a sum of two numbers that are almost the same.

Method. Near doubles: if two numbers are close, double the smaller one, then add the small difference.

Spot the near double. The numbers are very close together.
36,000 is 1,000 more than 35,000, so 36,000 = 35,000 + 1,000
Double 35,000. Double each part of the number, then add.
double 30,000 = 60,000
double 5,000 = 10,000
60,000 + 10,000 = 70,000
Add the extra 1,000. 36,000 is 1,000 more than 35,000, so the total is 1,000 more than the double.
70,000 + 1,000 = 71,000

Check. Use the opposite operation: 71,000 − 36,000 = 35,000. ✓

Watch out. Double the SMALLER number and add the difference. Do not add it to the bigger number.

Worked example 2.3.13

Work out 100,000 − 96,250 by counting up.

Key words. The difference between two numbers is how far apart they are. Counting up is a good way to find a difference when the numbers are close.

Method. To find the difference between two numbers, start at the smaller number and count up to the bigger one. Add up the jumps.

Start at the smaller number. The difference is how far we have to count up.
Start: 96,250. Finish: 100,000.
Count up in easy jumps. Jump to a round number first, then add the rest.
96,250 + 750 = 97,000
97,000 + 3,000 = 100,000
Add up the jumps. The total of all the jumps is the difference.
750 + 3,000 = 3,750

Check. Use addition: 96,250 + 3,750 = 100,000. ✓

Watch out. Add up the jumps. The answer is the size of the jumps, not the number you finish on.

Watch out

Compensating has two steps. People often do the first step (add 10,000) and then forget to fix the answer. 36,450 + 9,999 is not 46,450. Also be careful with the direction: when you take away too much, you must put some back. And with number facts, count the zeros: 75 + 25 = 100 but 75,000 + 25,000 = 100,000.

2.4

Estimating, rounding and checking

Year 5
Use rounding to check answers and decide the level of accuracy needed

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Estimating, rounding and checkingComing soon

Before you do a calculation, it helps to estimate the answer. An estimate is a good guess that is close to the real answer. To estimate, round the numbers, then do the easy calculation. In Chapter 1 you rounded to the nearest 10, 100, 1,000, 10,000 and 100,000. For 48,372 + 29,615, round to the nearest 1,000: 48,000 + 30,000 = 78,000. We write 48,372 + 29,615 ≈ 78,000. The sign ≈ means “is about equal to”.

48,372 is closer to 48,000 than to 49,000, so 48,372 rounds down to 48,000.

An estimate tells you if your answer is sensible. If your exact answer is far from your estimate, you have made a mistake somewhere.

How accurate does it need to be? Rounding to a bigger number (like the nearest 10,000) is quicker but less accurate. Rounding to a smaller number (like the nearest 100) takes longer but is closer. The right level depends on the question. For “about how many fans came?” the nearest 1,000 is fine. For “do I have enough money?” you may need to be much more careful.

Situation	Good level of accuracy
About how many fans were at the match?	nearest 1,000
About how far is it from London to Glasgow in metres?	nearest 10,000
Is my estimate close enough to check a column sum?	nearest 100 or 1,000
Do I have exactly enough money for the bill?	exact (no rounding)

Choose the level of accuracy to fit the question.

Checking with the inverse. Addition and subtraction are inverse (opposite) operations. They undo each other. If 45,368 + 27,594 = 72,962, then 72,962 − 27,594 should take us back to 45,368. The numbers make a fact family: two additions and two subtractions.

The whole is 72,962 and the two parts are 45,368 and 27,594.

The fact family: 45,368 + 27,594 = 72,962 27,594 + 45,368 = 72,962 72,962 − 27,594 = 45,368 72,962 − 45,368 = 27,594.

Missing numbers. To find a missing number in a sum like ? + 27,594 = 72,962, think of a bar model. The whole is 72,962. One part is 27,594. The other part is the missing number, so take the part away from the whole.

Key words. An estimate is a close answer. Inverse operations are opposites. Sensible means that the answer is the right sort of size. Accuracy means how close an answer is to the exact answer.

Key ideas

To estimate, round each number (to the nearest 10, 100, 1,000, 10,000 or 100,000), then do the easy calculation.
≈ means “is about equal to”.
Choose the level of accuracy to fit the question. If the estimate is close to a limit, use a closer estimate or the exact answer.
An exact answer should be close to the estimate. If not, check your working.
Addition and subtraction are inverses: if a + b = c, then c − b = a.
Use a bar model to find a missing number: find a part by subtracting, find a whole by adding.

Worked example 2.4.1

Estimate 48,372 + 29,615 by rounding to the nearest 1,000.

Key words. To estimate is to find an answer that is close, usually by rounding the numbers first. ≈ means “is about equal to”.

Method. Round each number to the nearest 1,000, then work out the easy calculation. The answer is only an estimate (a good guess), so it will not be exact.

Round 48,372 to the nearest 1,000. Look at the digit just to the right. 5 or more: round up. 4 or less: round down.
48,372 ≈ 48,000 (the hundreds digit is 3, so round down)
Round 29,615 to the nearest 1,000. Do the same for the next number.
29,615 ≈ 30,000 (the hundreds digit is 6, so round up)
Do the easy calculation. Rounded numbers are easy to add or subtract.
48,000 + 30,000 = 78,000

Check. The exact answer is 77,987. It is close to the estimate of 78,000. ✓

Watch out. An estimate is not the exact answer. Use the sign ≈ (about equal to).

Worked example 2.4.2

Estimate 652,480 − 187,950 by rounding to the nearest 10,000.

Key words. To estimate is to find an answer that is close, usually by rounding the numbers first. ≈ means “is about equal to”.

Method. Round each number to the nearest 10,000, then work out the easy calculation. The answer is only an estimate (a good guess), so it will not be exact.

Round 652,480 to the nearest 10,000. Look at the digit just to the right. 5 or more: round up. 4 or less: round down.
652,480 ≈ 650,000 (the thousands digit is 2, so round down)
Round 187,950 to the nearest 10,000. Do the same for the next number.
187,950 ≈ 190,000 (the thousands digit is 7, so round up)
Do the easy calculation. Rounded numbers are easy to add or subtract.
650,000 − 190,000 = 460,000

Check. The exact answer is 464,530. It is close to the estimate of 460,000. ✓

Watch out. An estimate is not the exact answer. Use the sign ≈ (about equal to).

Worked example 2.4.3

Estimate 24,380 + 8,720 + 15,950 by rounding each number to the nearest 1,000.

Key words. To estimate is to find an answer that is close, usually by rounding the numbers first. ≈ means “is about equal to”.

Method. Round each number to the nearest 1,000, then work out the easy calculation. The answer is only an estimate (a good guess), so it will not be exact.

Round 24,380 to the nearest 1,000. Look at the digit just to the right. 5 or more: round up. 4 or less: round down.
24,380 ≈ 24,000 (the hundreds digit is 3, so round down)
Round 8,720 to the nearest 1,000. Do the same for the next number.
8,720 ≈ 9,000 (the hundreds digit is 7, so round up)
Round 15,950 to the nearest 1,000. Do the same for the next number.
15,950 ≈ 16,000 (the hundreds digit is 9, so round up)
Do the easy calculation. Rounded numbers are easy to add or subtract.
24,000 + 9,000 + 16,000 = 49,000

Check. The exact answer is 49,050. It is close to the estimate of 49,000. ✓

Watch out. An estimate is not the exact answer. Use the sign ≈ (about equal to).

Worked example 2.4.4

Round 275,490 to the nearest 10,000.

Key words. To round a number is to change it to a nearby number that is easier to use. ≈ means “is about equal to”.

Method. Look at the digit to the right of the one you are rounding to. If it is 5 or more, round up. If it is 4 or less, round down.

Find the two multiples of 10,000 that 275,490 is between. Rounding to the nearest 10,000 means choosing the closest multiple of 10,000.
270,000 < 275,490 < 280,000
Look at the thousands digit (the digit just to the right of the ten thousands column). This digit tells us if we are past halfway.
The thousands digit is 5.
Decide: round up or round down? 5 or more: round up. 4 or less: round down.
5 is 5 or more, so round up.
Write the rounded number. Every digit to the right of the ten thousands column becomes a zero.
275,490 ≈ 280,000 (to the nearest 10,000)

Check. Distances: 275,490 − 270,000 = 5,490 and 280,000 − 275,490 = 4,510. 280,000 is the nearer one. ✓

Watch out. Only the digit just to the right decides. Do not look at digits further along.

Worked example 2.4.5

Sofia has £7,500 to spend. She wants tables for £2,540, chairs for £3,560 and a whiteboard for £1,390. She rounds each price to the nearest £1,000 and says, “I do not have enough.” Is Sofia right?

Method. Estimate first, but be careful: if the estimate is close to the amount we have, we must work out the exact answer.

Round each price to the nearest £1,000. This is quick, but it may not be accurate enough.
Tables: £2,540 ≈ £3,000
Chairs: £3,560 ≈ £4,000
Whiteboard: £1,390 ≈ £1,000
£3,000 + £4,000 + £1,000 = £8,000
Compare with Sofia’s £7,500. £8,000 is more than £7,500, so this estimate says there is not enough. But the estimate is £500 over, which is a lot of rounding error.
£8,000 > £7,500
Round to the nearest £100 for a closer estimate. Rounding to a smaller place value gives a more accurate answer.
Tables: £2,540 ≈ £2,500
Chairs: £3,560 ≈ £3,600
Whiteboard: £1,390 ≈ £1,400
£2,500 + £3,600 + £1,400 = £7,500
Work out the exact total. The estimate is very close to £7,500, so only the exact answer can decide.
£2,540 + £3,560 + £1,390 = £7,490
Th H T O
2 5 4 0
+ 3 5 6 0
+ 1 3 9 0
7 4 9 0
1 1
Decide. £7,490 is less than £7,500.
Sofia has enough. The total is £7,490, so £10 is left over.

Check. £7,490 + £10 = £7,500. ✓

Watch out. A rough estimate cannot answer a question when the numbers are close. Use a more accurate estimate, or the exact answer.

Worked example 2.4.6

Zainab estimates 36,640 + 28,580 by rounding each number to the nearest 1,000. She gets 66,000. Is the exact answer more or less than 66,000? Explain how you know without adding.

Method. Look at what happened to each number when it was rounded. Did it go up or down? Then think about what that does to the answer.

Compare each number with its rounded number. If a number was rounded up, the rounded number is bigger than the real one.
36,640 was rounded up to 37,000 (360 more). This makes the answer bigger.
28,580 was rounded up to 29,000 (420 more). This makes the answer bigger.
Decide what that means for the estimate. Both changes push the answer the same way, so we can be sure.
The estimate (66,000) is more than the exact answer, so the exact answer is less than 66,000.
Check with the exact answer.
36,640 + 28,580 = 65,220
TTh Th H T O
3 6 6 4 0
+ 2 8 5 8 0
6 5 2 2 0
1 1 1

Check. 65,220 is less than 66,000. ✓

Watch out. When you add, rounding up makes the answer too big. When you subtract, rounding the first number up makes the answer too big, but rounding the second number up makes it too small.

Worked example 2.4.7

Jack says that 64,205 − 38,790 = 34,595. Use an estimate to show that his answer is not sensible. Then find the correct answer.

Method. Estimate the answer by rounding. If Jack’s answer is nowhere near the estimate, it cannot be right.

Round the numbers. Round to the nearest 1,000 so the calculation is easy.
64,205 ≈ 64,000 (the hundreds digit is 2, so round down)
38,790 ≈ 39,000 (the hundreds digit is 7, so round up)
Work out the estimate.
64,000 − 39,000 = 25,000
Compare with Jack’s answer. Jack got 34,595. The estimate says the answer should be close to 25,000.
34,595 is a long way from 25,000, so it is not sensible.
Find the correct answer. Use the column method carefully.
TTh Th H T O
56 134 112 100 5
− 3 8 7 9 0
2 5 4 1 5

64,205 − 38,790 = 25,415

Check. The correct answer 25,415 is close to the estimate 25,000. ✓

Watch out. Always ask: is my answer about the right size? An estimate takes a few seconds and can catch big mistakes.

Worked example 2.4.8

Sam says that 45,368 + 27,594 = 72,962. Use the inverse operation to check if he is right.

Key words. Inverse operations are opposites. Addition is the inverse of subtraction, and subtraction is the inverse of addition.

Method. Use the inverse (opposite) operation to check. Addition and subtraction undo each other. If you go back, you should reach the number you started with.

Write Sam’s answer as an inverse calculation. The opposite operation should take us back to the first number.
45,368 + 27,594 = 72,962 so 72,962 − 27,594 should be 45,368
Work it out. Use the column method.
72,962 − 27,594 = 45,368
TTh Th H T O
67 122 89 156 122
− 2 7 5 9 4
4 5 3 6 8
Compare. We should get 45,368.
We got 45,368. 45,368 = 45,368, so Sam is correct.

Check. The correct answer is 72,962. Sam was right. ✓

Watch out. The inverse of + is −. The inverse of − is +. If the check does not work, one of the calculations has a mistake.

Worked example 2.4.9

Priya says that 81,305 − 36,749 = 44,666. Use the inverse operation to check if she is right.

Key words. Inverse operations are opposites. Addition is the inverse of subtraction, and subtraction is the inverse of addition.

Method. Use the inverse (opposite) operation to check. Addition and subtraction undo each other. If you go back, you should reach the number you started with.

Write Priya’s answer as an inverse calculation. The opposite operation should take us back to the first number.
81,305 − 36,749 = 44,666 so 44,666 + 36,749 should be 81,305
Work it out. Use the column method.
44,666 + 36,749 = 81,415
TTh Th H T O
4 4 6 6 6
+ 3 6 7 4 9
8 1 4 1 5
1 1 1 1
Compare. We should get 81,305.
We got 81,415. 81,415 is not 81,305, so Priya is wrong.
Find the correct answer. Do the original calculation again.
81,305 − 36,749 = 44,556
TTh Th H T O
78 101 123 90 155
− 3 6 7 4 9
4 4 5 5 6

Check. The correct answer is 44,556. 44,556 + 36,749 = 81,305. ✓

Watch out. The inverse of + is −. The inverse of − is +. If the check does not work, one of the calculations has a mistake.

Worked example 2.4.10

Find the missing number. ? + 27,594 = 72,962

Method. Draw a bar model, then use the inverse (opposite) operation to find the missing number.

Look at the number sentence. The box stands for the number we want to find.
? + 27,594 = 72,962
Draw a bar model. The whole is the total, and we know one part. The other part is the missing number.
Choose the operation. To find a missing part, take the part we know away from the whole.
72,962 − 27,594
Work it out. Use the column method if the numbers are big.
72,962 − 27,594 = 45,368
TTh Th H T O
67 122 89 156 122
− 2 7 5 9 4
4 5 3 6 8
Check in the number sentence. Put the answer back in the box.
45,368 + 27,594 = 72,962 ✓

Check. Read the whole number sentence with your answer in the box. It must be true. ✓

Watch out. The missing number is not always the last number. Think about which number is the whole, then decide whether to add or subtract.

Worked example 2.4.11

Find the missing number. 500,000 − ? = 284,615

Method. Draw a bar model, then use the inverse (opposite) operation to find the missing number.

Look at the number sentence. The box stands for the number we want to find.
500,000 − ? = 284,615
Draw a bar model. The whole is the first number. The missing number is one of the parts.
Choose the operation. To find a missing part, take the part we know away from the whole.
500,000 − 284,615
Work it out. Use the column method if the numbers are big.
500,000 − 284,615 = 215,385
HTh TTh Th H T O
45 90 90 90 90 100
− 2 8 4 6 1 5
2 1 5 3 8 5
Check in the number sentence. Put the answer back in the box.
500,000 − 215,385 = 284,615 ✓

Check. Read the whole number sentence with your answer in the box. It must be true. ✓

Watch out. The missing number is not always the last number. Think about which number is the whole, then decide whether to add or subtract.

Watch out

An estimate is not the exact answer, so do not expect it to match exactly. It should only be close. When you check with the inverse, use the opposite sign: check an addition by subtracting, and check a subtraction by adding. And when a total is close to a limit (such as a budget), rounding a lot can give the wrong decision.

2.5

Bar models and multi-step problems

Year 5
Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Bar models and multi-step problemsComing soon

A bar model is a picture that helps you see what a word problem is asking. You draw a bar, label it with the numbers you know, and put a question mark on the number you want. There are two main kinds of bar model.

Part-whole bar models show a whole that is made of parts. A town has 84,560 adults and 31,275 children. The whole is the number of people. The parts are the adults and the children. If you know the parts, add to find the whole. If you know the whole and one part, subtract to find the other part.

Part-whole: the two parts (adults and children) make the whole (all the people). Whole = 84,560 + 31,275.

Comparison bar models show two amounts side by side. The longer bar is the bigger amount. The dashed part is the difference: how much more the bigger amount is. Ben has 52,350 stamps and Cara has 38,975. To find how many more Ben has, subtract: 52,350 − 38,975.

Comparison: Ben has more than Cara. The dashed part is the difference.

Which operation, and why? The words in a problem are clues, but you must think about what the bar model shows.

What the problem says	What the bars show	Operation
altogether, total, in all	parts to make a whole	add
left, remaining, still needed	whole and one part, find the other part	subtract
how many more, the difference	two bars side by side, find the gap	subtract
5 more than (find the bigger amount)	smaller bar plus the gap	add
5 fewer than (find the smaller amount)	bigger bar minus the gap	subtract

The words “more” and “fewer” alone do not tell you whether to add or subtract. Look at the bars.

Multi-step problems. Many problems need two, three or more calculations. The answer to one step helps you with the next. Use these steps every time:

Read the problem twice. What do you know? What do you need to find?
Draw a bar model.
Plan your steps. What do you find first?
Calculate with a mental or written method. Choose the one that suits the numbers.
Check that your answer makes sense and that it answers the question that was asked.

Key words. A bar model is a drawing of the numbers in a problem. The whole is the total. A part is one piece of the whole. The difference is how much bigger one amount is than another. A multi-step problem needs more than one calculation.

Key ideas

Draw a bar model. Put a question mark on the number you want.
Part-whole: parts added together make the whole. Whole − part = other part.
Comparison: the dashed gap is the difference. Bigger − smaller = difference.
Bigger = smaller + difference. Smaller = bigger − difference.
Plan your steps before you calculate. Do not stop after step 1.
Check your answer by using the inverse operation, and ask: does it make sense?

Worked example 2.5.1

A town has 84,560 adults and 31,275 children. How many people live in the town?

Method. Draw a bar model to see what is happening. Decide whether to add or subtract, then use a written method.

Draw a bar model. A picture shows what we know and what we need to find.
Choose the operation. We know both parts and want the whole, so we add.
84,560 + 31,275
Work it out. Use the column method because the numbers are big.
84,560 + 31,275 = 115,835
HTh TTh Th H T O
8 4 5 6 0
+ 3 1 2 7 5
1 1 5 8 3 5
1 1
Write the answer.
115,835 people

Check. Use the opposite operation: 115,835 − 31,275 = 84,560. ✓

Watch out. Read the question again at the end. Does your answer make sense, and does it answer what was asked?

Worked example 2.5.2

A stadium has 350,000 tickets for a tournament. 218,750 tickets have been sold. How many tickets have not been sold?

Method. Draw a bar model to see what is happening. Decide whether to add or subtract, then use a written method.

Draw a bar model. A picture shows what we know and what we need to find.
Choose the operation. We know the whole and one part, and want the other part, so we subtract.
350,000 − 218,750
Work it out. Use the column method because the numbers are big.
350,000 − 218,750 = 131,250
HTh TTh Th H T O
3 45 90 90 100 0
− 2 1 8 7 5 0
1 3 1 2 5 0
Write the answer.
131,250 tickets

Check. Use the opposite operation: 131,250 + 218,750 = 350,000. ✓

Watch out. Read the question again at the end. Does your answer make sense, and does it answer what was asked?

Worked example 2.5.3

Ben has 52,350 stamps. Cara has 38,975 stamps. How many more stamps does Ben have than Cara?

Key words. A comparison bar model shows two amounts side by side. The difference is how much longer one bar is than the other.

Method. Draw a comparison bar model. The longer bar is the bigger amount. The dashed part is the difference. Then decide whether to add or subtract.

Draw a comparison bar model. The two bars let us see which amount is bigger and what the difference is.
Choose the operation. We want the difference between the two amounts, so we take the smaller from the bigger.
52,350 − 38,975
Work it out. Use the column method because the numbers are big.
52,350 − 38,975 = 13,375
TTh Th H T O
45 112 123 145 100
− 3 8 9 7 5
1 3 3 7 5
Write the answer.
13,375 stamps

Check. Use the opposite operation: 38,975 + 13,375 = 52,350. ✓

Watch out. “More than” does not always mean add. Draw the bars first, then think about which bar you are finding.

Worked example 2.5.4

Ben has 38,975 stamps. Cara has 14,600 more stamps than Ben. How many stamps does Cara have?

Key words. A comparison bar model shows two amounts side by side. The difference is how much longer one bar is than the other.

Method. Draw a comparison bar model. The longer bar is the bigger amount. The dashed part is the difference. Then decide whether to add or subtract.

Draw a comparison bar model. The two bars let us see which amount is bigger and what the difference is.
Choose the operation. Cara has 14,600 more than Ben, so we add.
38,975 + 14,600
Work it out. Use the column method because the numbers are big.
38,975 + 14,600 = 53,575
TTh Th H T O
3 8 9 7 5
+ 1 4 6 0 0
5 3 5 7 5
1 1
Write the answer.
53,575 stamps

Check. Use the opposite operation: 53,575 − 14,600 = 38,975. ✓

Watch out. “More than” does not always mean add. Draw the bars first, then think about which bar you are finding.

Worked example 2.5.5

A shop has 50,000 bottles of water. It sells 12,650 bottles in May and 18,475 bottles in June. How many bottles are left?

Method. This is a two-step problem. First add up everything that was used. Then take that total away from the amount at the start.

Draw a bar model. The whole bar is 50,000. It is split into the two amounts sold and the part that is left.
Step 1: add the two amounts. Put the two parts together.
12,650 + 18,475 = 31,125
TTh Th H T O
1 2 6 5 0
+ 1 8 4 7 5
3 1 1 2 5
1 1 1
Step 2: take the total away from the start. What is left is the whole minus the parts that were used.
50,000 − 31,125 = 18,875
TTh Th H T O
45 90 90 90 100
− 3 1 1 2 5
1 8 8 7 5
Write the answer.
18,875 bottles

Check. Add everything back: 12,650 + 18,475 + 18,875 = 50,000, which is the start amount. ✓

Watch out. Do not take away just one of the amounts. Both amounts were used, so take away the total of both.

Worked example 2.5.6

Mia scores 24,350 points in a game. Noah scores 8,900 more points than Mia. How many points do they score altogether?

Method. This is a two-step problem. First use the comparison to find the second amount. Then add the two amounts to find the total.

Draw a comparison bar model. The two bars show both amounts. The dashed part is the difference, 8,900.
Step 1: find the amount for Noah. Noah has 8,900 more, so we add.
24,350 + 8,900 = 33,250
TTh Th H T O
2 4 3 5 0
+ 8 9 0 0
3 3 2 5 0
1 1
Step 2: find the total. Add the two amounts: Mia and Noah.
24,350 + 33,250 = 57,600
TTh Th H T O
2 4 3 5 0
+ 3 3 2 5 0
5 7 6 0 0
1
Write the answer.
57,600 points

Check. The difference between the two amounts should be 8,900: 33,250 − 24,350 = 8,900. ✓

Watch out. A common mistake is to stop after step 1. The question asks for the total of BOTH amounts.

Worked example 2.5.7

A charity wants to raise £30,000. In May it raises £12,450. In June it raises £13,680. How much more does it need?

Method. This is a two-step problem. First find the total. Then compare it with the other amount by finding the difference.

Step 1: find the total. Put the two amounts together.
£12,450 + £13,680 = £26,130
TTh Th H T O
1 2 4 5 0
+ 1 3 6 8 0
2 6 1 3 0
1 1
Draw a comparison bar model. The dashed part is the difference we want.
Step 2: find the difference. Take the smaller amount from the bigger one.
£30,000 − £26,130 = £3,870
TTh Th H T O
23 90 90 100 0
− 2 6 1 3 0
3 8 7 0
Write the answer.
£3,870

Check. Add the difference to the smaller amount: £26,130 + £3,870 = £30,000. ✓

Watch out. Take the SMALLER amount from the BIGGER amount to find the difference.

Worked example 2.5.8

There are 65,400 fans in a stadium. At half time 28,750 fans leave and 17,300 new fans arrive. How many fans are in the stadium now?

Method. This is a two-step problem. Follow the story in order. First take away the amount that goes out. Then add the amount that comes in.

Draw what happens. A number line shows the story in order: first it goes down, then it goes up.
Step 1: take away. 28,750 leave the stadium, so we subtract.
65,400 − 28,750 = 36,650
TTh Th H T O
56 145 134 100 0
− 2 8 7 5 0
3 6 6 5 0
Step 2: add. Now 17,300 arrive, so we add to the new total.
36,650 + 17,300 = 53,950
TTh Th H T O
3 6 6 5 0
+ 1 7 3 0 0
5 3 9 5 0
1
Write the answer.
53,950 fans

Check. Work backwards: 53,950 − 17,300 = 36,650 and 36,650 + 28,750 = 65,400. ✓

Watch out. Follow the story in order. The first thing that happens is the first calculation.

Worked example 2.5.9

Tom had some money. He spent £2,450 on a bike and £685 on a helmet. He now has £3,865 left. How much money did Tom have at the start?

Method. Draw a bar model. The whole is bigger than every part, so add the parts together to find it.

Draw a bar model. The whole is what Tom had at the start. The parts are the bike, the helmet and the money that is left.
Choose the operation. We know all the parts and want the whole, so we add.
£2,450 + £685 + £3,865
Work it out. Use the column method because the numbers are big.
£2,450 + £685 + £3,865 = £7,000
Th H T O
2 4 5 0
+ 6 8 5
+ 3 8 6 5
7 0 0 0
2 2 1
Write the answer.
£7,000

Check. Take the parts away again: £7,000 − £2,450 − £685 = £3,865. ✓

Watch out. This is a “working backwards” problem. The whole is bigger than every part, so you add, not subtract.

Worked example 2.5.10

Ben has 38,250 stamps. Amira has 12,500 more stamps than Ben. Cara has 3,200 fewer stamps than Amira. How many stamps do the three children have altogether?

Method. This is a three-step problem. Work out each unknown amount in order, then add all three at the end.

Draw the bars. Amira is longer than Ben. Cara is shorter than Amira.
Step 1: find how many Amira has. Amira has 12,500 more than Ben, so we add.
38,250 + 12,500 = 50,750
TTh Th H T O
3 8 2 5 0
+ 1 2 5 0 0
5 0 7 5 0
1
Step 2: find how many Cara has. Cara has 3,200 fewer than Amira, so we subtract.
50,750 − 3,200 = 47,550
TTh Th H T O
45 100 7 5 0
− 3 2 0 0
4 7 5 5 0
Step 3: add all three amounts. The question asks for the total of all three.
38,250 + 50,750 + 47,550 = 136,550
HTh TTh Th H T O
3 8 2 5 0
+ 5 0 7 5 0
+ 4 7 5 5 0
1 3 6 5 5 0
1 1 1 1
Write the answer.
136,550 stamps

Check. Check each comparison: 50,750 − 38,250 = 12,500 ✓ and 3,200 is the difference between Amira and Cara. ✓

Watch out. Do not stop early. Each step gives a number you need for the next one, and the last step answers the question.

Watch out

The word “fewer” does not always mean subtract, and “more” does not always mean add. In “Cara has 14,600 more than Ben” you add to find Cara, but you subtract to find the difference. Draw the bars first and decide which bar you are finding. Also remember that the answer to step 1 is not the final answer.

2.6

Missing digits and number puzzles

Year 5
Use the inverse operation to find missing numbers and digits, Reason about calculations

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Missing digits and number puzzlesComing soon

In a missing digit puzzle some digits in a column calculation are hidden. Each box stands for one digit from 0 to 9. You can find the digits with logic, and there is only one right answer.

Addition puzzles. Start with the ones column on the right, like you always do. If the missing digit is in the answer, just add. If it is in one of the numbers, ask: “what do I add to the digit I know to get a number that ends in the answer digit?” Then think about the carry, because it matters in the next column.

	TTh	Th	H	T	O
	4	5	?	6	8
+	2	7	5	?	4
	7	2	9	6	?

Three digits are hidden. Each box stands for one digit. The ones column has a hidden digit in the answer, so we can add it straight away.

Subtraction puzzles. Turn a subtraction into an addition. If 84,270 − 51,835 = 32,435, then 51,835 + 32,435 = 84,270. The inverse sum has the same digits, and you can solve it from the right.

	TTh	Th	H	T	O
	8	4	?	7	0
−	5	?	8	3	5
	3	2	4	3	?

A subtraction puzzle. Turn it into 51,835 + 32,435 = 84,270 to solve it.

Balance equations. The equals sign (=) means “is the same as”. It does not always have the answer on its right. In 63,500 + ? = 100,000 − 14,500, work out the side with no box first (85,500). Then find the missing number: 85,500 − 63,500 = 22,000.

Reasoning puzzles. Number-card puzzles ask you to think about the value of each column. A digit in a big column matters more than a digit in a small column. For a big total, put big digits in big columns. For a small difference, make the numbers as close as you can.

Key words. A digit is one of 0 to 9. An equation says that two amounts are equal. The inverse of an operation undoes it.

Key ideas

Always start with the ones column and work to the left.
Remember the carry. A missing digit must make the column total end in the right digit.
Turn a subtraction puzzle into an addition with the inverse: top − bottom = answer means bottom + answer = top.
The = sign means “is the same as”. Work out the side with no box first.
For a big total put big digits in big columns. For a small difference make the numbers close.
Check by doing the completed calculation.

Worked example 2.6.1

Find the missing digits.

	TTh	Th	H	T	O
	4	5	?	6	8
+	2	7	5	?	4
	7	2	9	6	?

Key words. A digit is one of 0 to 9. The ones column is the column on the right.

Method. Work from the ones column on the right. In each column use the digits you know, and the carry from the column before, to find the missing digit.

Find the missing digit in the ones column of the answer. We know the digits we are adding, so we add them (and the carry).
8 + 4 = 12. The missing digit is 2. Carry 1 to the tens column.
TTh Th H T O
4 5 ? 6 8
+ 2 7 5 ? 4
7 2 9 6 2
Find the missing digit in the tens column of the second number. The answer digit is 6, so the column total must end in 6. What do we add to the digits we know?
? + 6 + 1 must end in 6.
Count on from 7 to a number that ends in 6: 7 + 9 = 16.
The missing digit is 9. Carry 1 to the hundreds column.
TTh Th H T O
4 5 ? 6 8
+ 2 7 5 9 4
7 2 9 6 2
Find the missing digit in the hundreds column of the first number. The answer digit is 9, so the column total must end in 9. What do we add to the digits we know?
? + 5 + 1 must end in 9.
Count on from 6 to a number that ends in 9: 6 + 3 = 9.
The missing digit is 3.
TTh Th H T O
4 5 3 6 8
+ 2 7 5 9 4
7 2 9 6 2
Look at the thousands column. Nothing is hidden here, but we need to know the carry for the next column.
5 + 7 = 12. Write 2. Carry 1 to the ten thousands column.
TTh Th H T O
4 5 3 6 8
+ 2 7 5 9 4
7 2 9 6 2
Look at the ten thousands column. Nothing is hidden here.
4 + 2 + 1 = 7. Write 7.
TTh Th H T O
4 5 3 6 8
+ 2 7 5 9 4
7 2 9 6 2
Write the completed calculation. Put the missing digits back and read the whole calculation.
TTh Th H T O
4 5 3 6 8
+ 2 7 5 9 4
7 2 9 6 2
1 1 1

45,368 + 27,594 = 72,962
answer, ones digit: 2
second number, tens digit: 9
first number, hundreds digit: 3

Check. Add again: 45,368 + 27,594 = 72,962. ✓

Watch out. Do not forget the carry. A missing digit must make the column total end in the right digit. Then the carry goes to the next column.

Worked example 2.6.2

Find the missing digits.

	HTh	TTh	Th	H	T	O
	?	5	8	7	6	4
+	2	4	?	6	8	9
	6	?	6	4	?	3

Key words. A digit is one of 0 to 9. The ones column is the column on the right.

Method. Work from the ones column on the right. In each column use the digits you know, and the carry from the column before, to find the missing digit.

Look at the ones column. Nothing is hidden here, but we need to know the carry for the next column.
4 + 9 = 13. Write 3. Carry 1 to the tens column.
Find the missing digit in the tens column of the answer. We know the digits we are adding, so we add them (and the carry).
6 + 8 + 1 = 15. The missing digit is 5. Carry 1 to the hundreds column.
Look at the hundreds column. Nothing is hidden here, but we need to know the carry for the next column.
7 + 6 + 1 = 14. Write 4. Carry 1 to the thousands column.
Find the missing digit in the thousands column of the second number. The answer digit is 6, so the column total must end in 6. What do we add to the digits we know?
? + 8 + 1 must end in 6.
Count on from 9 to a number that ends in 6: 9 + 7 = 16.
The missing digit is 7. Carry 1 to the ten thousands column.
Find the missing digit in the ten thousands column of the answer. We know the digits we are adding, so we add them (and the carry).
5 + 4 + 1 = 10. The missing digit is 0. Carry 1 to the hundred thousands column.
Find the missing digit in the hundred thousands column of the first number. The answer digit is 6, so the column total must end in 6. What do we add to the digits we know?
? + 2 + 1 must end in 6.
Count on from 3 to a number that ends in 6: 3 + 3 = 6.
The missing digit is 3.
Write the completed calculation. Put the missing digits back and read the whole calculation.
HTh TTh Th H T O
3 5 8 7 6 4
+ 2 4 7 6 8 9
6 0 6 4 5 3
1 1 1 1 1

358,764 + 247,689 = 606,453
answer, tens digit: 5
second number, thousands digit: 7
answer, ten thousands digit: 0
first number, hundred thousands digit: 3

Check. Add again: 358,764 + 247,689 = 606,453. ✓

Watch out. Do not forget the carry. A missing digit must make the column total end in the right digit. Then the carry goes to the next column.

Worked example 2.6.3

Find the missing digits.

	TTh	Th	H	T	O
	8	4	?	7	0
−	5	?	8	3	5
	3	2	4	3	?

Key words. A digit is one of 0 to 9. The ones column is the column on the right.

Method. Use the inverse operation to turn the subtraction into an addition. Then work from the ones column on the right.

Turn the subtraction into an addition. Subtraction is the inverse of addition: the answer plus the number taken away makes the number we started with. Now we can work from the right, like an addition.
84,270 − 51,835 = 32,435 means 51,835 + 32,435 = 84,270
TTh Th H T O
5 ? 8 3 5
+ 3 2 4 3 ?
8 4 ? 7 0
Find the missing digit in the ones column of the answer. The answer digit is 0, so the column total must end in 0. What do we add to the digits we know?
? + 5 must end in 0.
Count on from 5 to a number that ends in 0: 5 + 5 = 10.
The missing digit is 5. Carry 1 to the tens column.
TTh Th H T O
5 ? 8 3 5
+ 3 2 4 3 5
8 4 ? 7 0
Look at the tens column. Nothing is hidden here.
3 + 3 + 1 = 7. Write 7.
TTh Th H T O
5 ? 8 3 5
+ 3 2 4 3 5
8 4 ? 7 0
Find the missing digit in the hundreds column of the top number. We know the digits we are adding, so we add them (and the carry).
8 + 4 = 12. The missing digit is 2. Carry 1 to the thousands column.
TTh Th H T O
5 ? 8 3 5
+ 3 2 4 3 5
8 4 2 7 0
Find the missing digit in the thousands column of the bottom number. The answer digit is 4, so the column total must end in 4. What do we add to the digits we know?
? + 2 + 1 must end in 4.
Count on from 3 to a number that ends in 4: 3 + 1 = 4.
The missing digit is 1.
TTh Th H T O
5 1 8 3 5
+ 3 2 4 3 5
8 4 2 7 0
Look at the ten thousands column. Nothing is hidden here.
5 + 3 = 8. Write 8.
TTh Th H T O
5 1 8 3 5
+ 3 2 4 3 5
8 4 2 7 0
Write the completed calculation. Put the missing digits back and read the whole calculation.
TTh Th H T O
8 34 122 67 100
− 5 1 8 3 5
3 2 4 3 5

84,270 − 51,835 = 32,435
answer, ones digit: 5
top number, hundreds digit: 2
bottom number, thousands digit: 1

Check. Subtract again with the completed numbers: 84,270 − 51,835 = 32,435. ✓

Watch out. Do not forget the carry. A missing digit must make the column total end in the right digit. Then the carry goes to the next column.

Worked example 2.6.4

Find the missing digits.

	HTh	TTh	Th	H	T	O
	6	0	0	0	?	0
−	1	?	4	5	6	8
	?	7	5	?	3	2

Key words. A digit is one of 0 to 9. The ones column is the column on the right.

Method. Use the inverse operation to turn the subtraction into an addition. Then work from the ones column on the right.

Turn the subtraction into an addition. Subtraction is the inverse of addition: the answer plus the number taken away makes the number we started with. Now we can work from the right, like an addition.
600,000 − 124,568 = 475,432 means 124,568 + 475,432 = 600,000
HTh TTh Th H T O
1 ? 4 5 6 8
+ ? 7 5 ? 3 2
6 0 0 0 ? 0
Look at the ones column. Nothing is hidden here, but we need to know the carry for the next column.
8 + 2 = 10. Write 0. Carry 1 to the tens column.
Find the missing digit in the tens column of the top number. We know the digits we are adding, so we add them (and the carry).
6 + 3 + 1 = 10. The missing digit is 0. Carry 1 to the hundreds column.
Find the missing digit in the hundreds column of the answer. The answer digit is 0, so the column total must end in 0. What do we add to the digits we know?
? + 5 + 1 must end in 0.
Count on from 6 to a number that ends in 0: 6 + 4 = 10.
The missing digit is 4. Carry 1 to the thousands column.
Look at the thousands column. Nothing is hidden here, but we need to know the carry for the next column.
4 + 5 + 1 = 10. Write 0. Carry 1 to the ten thousands column.
Find the missing digit in the ten thousands column of the bottom number. The answer digit is 0, so the column total must end in 0. What do we add to the digits we know?
? + 7 + 1 must end in 0.
Count on from 8 to a number that ends in 0: 8 + 2 = 10.
The missing digit is 2. Carry 1 to the hundred thousands column.
Find the missing digit in the hundred thousands column of the answer. The answer digit is 6, so the column total must end in 6. What do we add to the digits we know?
? + 1 + 1 must end in 6.
Count on from 2 to a number that ends in 6: 2 + 4 = 6.
The missing digit is 4.
Write the completed calculation. Put the missing digits back and read the whole calculation.
HTh TTh Th H T O
56 90 90 90 90 100
− 1 2 4 5 6 8
4 7 5 4 3 2

600,000 − 124,568 = 475,432
top number, tens digit: 0
answer, hundreds digit: 4
bottom number, ten thousands digit: 2
answer, hundred thousands digit: 4

Check. Subtract again with the completed numbers: 600,000 − 124,568 = 475,432. ✓

Watch out. Do not forget the carry. A missing digit must make the column total end in the right digit. Then the carry goes to the next column.

Worked example 2.6.5

Find the missing number. 63,500 + ? = 100,000 − 14,500

Key words. An equation says that two amounts are equal. The left-hand side and right-hand side are the amounts on either side of the = sign.

Method. Both sides of the equals sign must have the same value. Work out the side with no box first.

Look at the number sentence. The = sign means both sides are worth the same.
63,500 + ? = 100,000 − 14,500
Work out the right-hand side. It has no box, so we can work it out.
100,000 − 14,500 = 85,500
HTh TTh Th H T O
01 90 90 100 0 0
− 1 4 5 0 0
8 5 5 0 0
Find the missing number. To find a missing part, take the part we know away from the whole.
85,500 − 63,500
85,500 − 63,500 = 22,000
TTh Th H T O
8 5 5 0 0
− 6 3 5 0 0
2 2 0 0 0
Check. Put the answer in the box and work out both sides.
63,500 + 22,000 = 85,500 and the other side is 85,500. ✓

Check. Both sides give the same number. ✓

Watch out. The = sign does not mean “the answer is next”. It means “is the same as”, so a calculation can be on either side.

Worked example 2.6.6

What is the smallest digit that can go in the box so that 6,5?0 + 2,480 is more than 9,000?

Method. Find out how much each 1 in the box changes the total. Then work out which digit is the first to make the sum true.

Put 0 in the box and work out the sum. This gives the start of the list.
6,500 + 2,480 = 8,980
See what each extra 1 in the box does. The box is in the tens column, so each 1 more in the box makes the total 10 bigger.
0 → 8,980, 1 → 8,990, 2 → 9,000, …
Compare with 9,000. We need the sum to be more than 9,000. Check the digits next to the answer.
box = 2: 6,520 + 2,480 = 9,000 (not more than 9,000)
box = 3: 6,530 + 2,480 = 9,010 > 9,000
box = 4: 6,540 + 2,480 = 9,020 > 9,000
Write the answer.
The smallest digit is 3.

Check. With 3 in the box the sum is 9,010, which is more than 9,000. ✓

Watch out. Read the question carefully: “more than” does not include the limit itself. If the sum is exactly equal to the limit, that digit does not work.

Worked example 2.6.7

Use each of the digit cards 1, 2, 3, 4, 5 and 6 once to make two three-digit numbers. What is the smallest difference you can make?

Method. To make the difference small, the two numbers must be close. Make the hundreds digits as close as possible, then make the top number as small as it can be and the bottom number as big as it can be.

Choose the hundreds digits. The hundreds digits matter most. They should be next to each other in the list of digits.
Digits in order: 1, 2, 3, 4, 5, 6
Fill the other columns. The top number (the bigger one) gets the smallest digits that are left. The bottom number (the smaller one) gets the biggest digits that are left. Then the gap between them is as small as possible.
hundreds digits 2 and 1: 234 − 165 = 69
hundreds digits 3 and 2: 314 − 265 = 49
hundreds digits 4 and 3: 412 − 365 = 47
hundreds digits 5 and 4: 512 − 463 = 49
hundreds digits 6 and 5: 612 − 543 = 69
Find the smallest difference. Compare all the differences.
The smallest is 412 − 365 = 47

Check. Check: 365 + 47 = 412. ✓ (Every other way of using the cards gives a bigger difference.)

Watch out. Use every card exactly once. Check that your two numbers use all six different digits.

Watch out

A box in a number sentence is not always at the end, and the answer is not always on the right of the = sign. In a puzzle, do not guess: use the ones column and the carry. And when you have an answer, always put the digits back and check the whole calculation.

2.7

Tables, charts, timetables and bank statements

Year 5
Solve sum and difference problems using information in tables, bar charts, timetables and bank statements, Choose which method to use

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Tables, charts, timetables and bank statementsComing soon

Numbers in real life often come in tables, charts and timetables. Adding and subtracting helps you to find totals and differences. First read the information carefully. Then decide which numbers you need, which operation to use and which method is best.

Town	Population
Ashby	84,560
Brigg	126,400
Calder	47,285
Denby	91,750

A table of populations. To find the total of two towns, add. To compare two towns, find the difference.

Bar charts. Look at the scale on the side. Here every labelled line is 10,000 and there are 5 small steps between them, so each small step is 2,000. A bar that ends two small steps above the 30,000 line is 34,000. Do not just read the nearest number on the scale: count the small steps.

A bar chart of the number of people at a festival each day. The scale goes up in 10,000s with 5 small steps of 2,000 between them.

Timetables. Times are written in hours and minutes, with 60 minutes in an hour. To find how long a journey takes, count on from the start time to the end time: first to the next o’clock, then the rest.

Bus stop	Time
School	08:15
Park	08:32
Library	08:47
Station	09:05
Market	09:38

A bus timetable (24-hour clock). Park 08:32 to Station 09:05 is 28 minutes to 09:00, then 5 more, 33 minutes in all.

Bank statements. A bank statement shows money going in (like wages) and money going out (like rent). The balance is the money left in the account. Money in is added. Money out is subtracted.

What happened	Money in	Money out	Balance
Opening balance			£2,450
Wages	£1,865		£4,315
Rent		£950	£3,365
Gas bill		£185	£3,180
Gift	£250		£3,430

A bank statement. Each balance comes from the line above it.

Which method? Before you calculate, look at the numbers and ask yourself these questions.

Are the numbers friendly, or close to round numbers? Mental method (compensate, count up, number bonds).
Are the numbers big and messy with lots of carrying or exchanging? Column method.
Do I only need a rough idea? Estimate by rounding.
Do I need an exact answer, such as for money or a limit? Exact calculation.

Then check your answer with an estimate or the inverse. There is often more than one good method, but you should be able to explain why you chose yours.

Key words. A table shows information in rows and columns. A bar chart uses bars to show amounts. A timetable shows when things happen. A balance is the amount of money in an account.

Key ideas

Read the table or chart carefully. Find the numbers you need and note the units.
On a bar chart, work out what each small step on the scale is worth.
For a timetable, count on to the next o’clock, then add the rest. There are 60 minutes in an hour.
On a bank statement, money in adds and money out subtracts.
Choose a method that suits the numbers. Mental for friendly numbers, column for messy ones.
Check with an estimate or the inverse.

Worked example 2.7.1

The table shows the populations of four towns.

Town	Population
Ashby	84,560
Brigg	126,400
Calder	47,285
Denby	91,750

What is the total population of Ashby, Brigg and Calder?

Method. Write the numbers in columns with the same place values lined up. Add the ones first, then the tens, hundreds and so on. If a column makes 10 or more, carry to the next column.

Line up the digits in columns. Ones go under ones, tens under tens, hundreds under hundreds and so on, so every digit is added to digits of the same value. The numbers have different lengths, so line them up on the right.
HTh TTh Th H T O
8 4 5 6 0
+ 1 2 6 4 0 0
+ 4 7 2 8 5
Add the ones. Always start with the ones, on the right.
0 + 0 + 5 = 5
5 is less than 10, so write 5 in the ones column. There is nothing to carry.
Add the tens. Now the next column to the left.
6 + 0 + 8 = 14
14 tens is 1 hundred and 4 tens. Write 4 in the tens column and carry 1 to the hundreds column.
Add the hundreds. Now the next column to the left. Add the 1 we carried as well.
5 + 4 + 2 + 1 = 12 (the red digit is the carry)
12 hundreds is 1 thousand and 2 hundreds. Write 2 in the hundreds column and carry 1 to the thousands column.
Add the thousands. Now the next column to the left. Add the 1 we carried as well.
4 + 6 + 7 + 1 = 18 (the red digit is the carry)
18 thousands is 1 ten thousand and 8 thousands. Write 8 in the thousands column and carry 1 to the ten thousands column.
Add the ten thousands. Now the next column to the left. Add the 1 we carried as well.
8 + 2 + 4 + 1 = 15 (the red digit is the carry)
15 ten thousands is 1 hundred thousand and 5 ten thousands. Write 5 in the ten thousands column and carry 1 to the hundred thousands column.
Add the hundred thousands. Now the next column to the left. Add the 1 we carried as well.
1 + 1 = 2 (the red digit is the carry)
2 is less than 10, so write 2 in the hundred thousands column. There is nothing to carry.
Read the answer. Check that the answer has a sensible number of digits, then write it.
HTh TTh Th H T O
8 4 5 6 0
+ 1 2 6 4 0 0
+ 4 7 2 8 5
2 5 8 2 4 5
1 1 1 1

84,560 + 126,400 + 47,285 = 258,245

Check. Use the opposite (inverse) operation: 258,245 − 126,400 − 47,285 = 84,560. ✓
Estimate: 85,000 + 126,000 + 47,000 = 258,000 (rounding to the nearest 1,000). It is close to the exact answer. ✓

Watch out. Do not forget the carried digits. Write each one small under the answer line, and add it in the next column.

Worked example 2.7.2

Use the table of populations. How many more people live in Brigg than in Calder?

Key words. A comparison bar model shows two amounts side by side. The difference is how much longer one bar is than the other.

Method. Draw a comparison bar model. The longer bar is the bigger amount. The dashed part is the difference. Then decide whether to add or subtract.

Draw a comparison bar model. The two bars let us see which amount is bigger and what the difference is.
Choose the operation. We want the difference between the two amounts, so we take the smaller from the bigger.
126,400 − 47,285
Work it out. Use the column method because the numbers are big.
126,400 − 47,285 = 79,115
HTh TTh Th H T O
01 112 166 34 90 100
− 4 7 2 8 5
7 9 1 1 5
Write the answer.
79,115 people

Check. Use the opposite operation: 47,285 + 79,115 = 126,400. ✓

Watch out. “More than” does not always mean add. Draw the bars first, then think about which bar you are finding.

Worked example 2.7.3

The bar chart shows the number of people at a festival each day.

How many more people were there on Thursday than on Wednesday?

Method. Read the height of each bar using the scale. Then find the difference.

Read the scale. The labelled lines go up in 10,000s. There are 5 small steps between them, so each small step is 10,000 ÷ 5 = 2,000.
10,000 ÷ 5 = 2,000
Read the Thursday bar. It ends 3 small steps above the 40,000 line: 3 × 2,000 = 6,000.
40,000 + 6,000 = 46,000
Read the Wednesday bar. It ends 1 small step below the 30,000 line: 30,000 − 2,000.
30,000 − 2,000 = 28,000
Find the difference. Take the smaller number from the bigger number.
46,000 − 28,000 = 18,000
TTh Th H T O
34 166 0 0 0
− 2 8 0 0 0
1 8 0 0 0

Check. 28,000 + 18,000 = 46,000. ✓

Watch out. Do not read the nearest label on the scale. Count the small steps.

Worked example 2.7.4

Use the bar chart. How many people went to the festival on Monday, Tuesday and Wednesday altogether? (Read the bars: 34,000, 42,000 and 28,000.)

Method. Write the numbers in columns with the same place values lined up. Add the ones first, then the tens, hundreds and so on. If a column makes 10 or more, carry to the next column.

Line up the digits in columns. Ones go under ones, tens under tens, hundreds under hundreds and so on, so every digit is added to digits of the same value.
HTh TTh Th H T O
3 4 0 0 0
+ 4 2 0 0 0
+ 2 8 0 0 0
Add the ones. Always start with the ones, on the right.
0 + 0 + 0 = 0
0 is less than 10, so write 0 in the ones column. There is nothing to carry.
Add the tens. Now the next column to the left.
0 + 0 + 0 = 0
0 is less than 10, so write 0 in the tens column. There is nothing to carry.
Add the hundreds. Now the next column to the left.
0 + 0 + 0 = 0
0 is less than 10, so write 0 in the hundreds column. There is nothing to carry.
Add the thousands. Now the next column to the left.
4 + 2 + 8 = 14
14 thousands is 1 ten thousand and 4 thousands. Write 4 in the thousands column and carry 1 to the ten thousands column.
Add the ten thousands. Now the next column to the left. Add the 1 we carried as well.
3 + 4 + 2 + 1 = 10 (the red digit is the carry)
10 ten thousands is 1 hundred thousand and 0 ten thousands. Write 0 in the ten thousands column and carry 1 to the hundred thousands column.
Write the carried digit. There are no digits left in the hundred thousands column, but we carried 1 into it.
There is only one digit here, so write 1 in the hundred thousands column.
Read the answer. Check that the answer has a sensible number of digits, then write it.
HTh TTh Th H T O
3 4 0 0 0
+ 4 2 0 0 0
+ 2 8 0 0 0
1 0 4 0 0 0
1 1

34,000 + 42,000 + 28,000 = 104,000

Check. Use the opposite (inverse) operation: 104,000 − 42,000 − 28,000 = 34,000. ✓
Estimate: 34,000 + 42,000 + 28,000 = 104,000 (rounding to the nearest 1,000). It is close to the exact answer. ✓

Watch out. Do not forget the carried digits. Write each one small under the answer line, and add it in the next column.

Worked example 2.7.5

Mr Ali’s bank statement is shown.

What happened	Money in	Money out	Balance
Opening balance			£2,450
Wages	£1,865		?
Rent		£950	?
Gas bill		£185	?
Gift	£250		?

Work out the balance after each line.

Key words. A bank statement lists the money going in and out of an account. The balance is the amount of money in the account.

Method. The balance is the money in the account. Start with the opening balance. Add money that comes in and take away money that goes out, one line at a time.

Start with the opening balance. This is the first number.
Opening balance = £2,450
Wages come in. Money in is added to the balance. Money out is taken away.
2,450 + 1,865 = 4,315
Th H T O
2 4 5 0
+ 1 8 6 5
4 3 1 5
1 1
Rent goes out. Use the new balance from the step before.
4,315 − 950 = 3,365
Th H T O
34 123 111 5
− 9 5 0
3 3 6 5
Gas bill goes out. Use the new balance from the step before.
3,365 − 185 = 3,180
Th H T O
3 23 166 5
− 1 8 5
3 1 8 0
A gift comes in. Use the new balance from the step before.
3,180 + 250 = 3,430
Th H T O
3 1 8 0
+ 2 5 0
3 4 3 0
1
Write the final balance. It is the last balance.
£3,430

Check. Check the total: £2,450 + money in − money out = £2,450 + £2,115 − £1,135 = £3,430. ✓

Watch out. Money out (a payment) makes the balance smaller. Money in (a deposit or wages) makes it bigger.

Worked example 2.7.6

Use the bus timetable.

Bus stop	Time
School	08:15
Park	08:32
Library	08:47
Station	09:05
Market	09:38

How many minutes does the bus take to go from Park to Station?

Method. Count on from the earlier time to the later time. Jump to the next whole hour first, then add the rest. Add up the jumps.

Write the two times. The journey starts at the earlier time and ends at the later time.
Start 08:32, finish 09:05
Count on in jumps. Jump to the next o’clock first, because it is easy to count from there.
08:32 + 28 minutes = 09:00
09:00 + 5 minutes = 09:05
Add up the jumps. The total of the jumps is the number of minutes.
28 + 5 = 33 minutes

Check. 08:32 + 33 minutes = 09:05. ✓

Watch out. There are 60 minutes in an hour, not 100. Do not use column subtraction on clock times.

Worked example 2.7.7

A pilot flies a route in four stages.

Stage	Distance (km)
London to New York	5,570
New York to Los Angeles	3,940
Los Angeles to Tokyo	8,830
Tokyo to Delhi	5,840

Dev has flown the first two stages. How far does he still have to fly?

Method. Plan the steps. Find the total distance, find the distance already flown, then subtract. (There is a quicker way too: add the stages still to fly.)

Add all four stages. This gives the whole journey.
5,570 km + 3,940 km + 8,830 km + 5,840 km = 24,180 km
TTh Th H T O
5 5 7 0
+ 3 9 4 0
+ 8 8 3 0
+ 5 8 4 0
2 4 1 8 0
2 3 1
Add the first two stages. This is the distance Dev has already flown.
5,570 km + 3,940 km = 9,510 km
Th H T O
5 5 7 0
+ 3 9 4 0
9 5 1 0
1 1
Subtract. Whole journey minus the distance flown is the distance left.
24,180 km − 9,510 km = 14,670 km
TTh Th H T O
12 134 111 8 0
− 9 5 1 0
1 4 6 7 0
Check the quicker way. The distance left is also the last two stages added together.
8,830 km + 5,840 km = 14,670 km
TTh Th H T O
8 8 3 0
+ 5 8 4 0
1 4 6 7 0
1 1

Check. 9,510 + 14,670 = 24,180 km. ✓

Watch out. Both ways give the same answer. Choose the one that needs fewer steps.

Worked example 2.7.8

Choose the best method for each calculation: mental or column. Then work out the answer and say why you chose that method.
4,999 + 3,001 45,368 + 27,594 70,000 − 69,994 306,040 − 148,975

Method. For each calculation, look at the numbers first. Friendly numbers suit a mental method. Messy numbers suit the column method.

4,999 + 3,001. Mental method. 3,001 + 5,000 = 8,001, then take 1 away.
8,000
45,368 + 27,594. Column method. Lots of carrying, so a written method is safer.
72,962
70,000 − 69,994. Mental method. The numbers are very close, so count up: 69,994 + 6 = 70,000.
6
306,040 − 148,975. Column method. Zeros and exchanges, so use columns.
157,065
Check the choices. There can be more than one good method, but you must be able to explain why you chose it.
Each answer can be checked with the inverse operation.

Check. 8,000 − 3,001 = 4,999 ✓; 72,962 − 27,594 = 45,368 ✓; 69,994 + 6 = 70,000 ✓; 157,065 + 148,975 = 306,040. ✓

Watch out. Do not use a column method for 4,999 + 3,001 just because it is an addition. A mental method is quicker and less likely to go wrong.

Watch out

Read bar charts carefully: a bar that is between two labelled lines is not equal to the nearest label. Count the small steps. On a bank statement do not add money out: money out is subtracted from the balance. And do not use column subtraction for clock times, because there are 60 minutes in an hour, not 100.

Exercise 2G Tables, charts, timetables and bank statements

Non-calculator

FThe table shows the populations of four towns. Use it to answer the questions.

Town	Population
Ashby	84,560
Brigg	126,400
Calder	47,285
Denby	91,750

(a)What is the total population of Ashby and Calder?

Method. Write the numbers in columns with the same place values lined up. Add the ones first, then the tens, hundreds and so on. If a column makes 10 or more, carry to the next column.

Line up the digits in columns. Ones go under ones, tens under tens, hundreds under hundreds and so on, so every digit is added to digits of the same value.
HTh TTh Th H T O
8 4 5 6 0
+ 4 7 2 8 5
Add the ones. Always start with the ones, on the right.
0 + 5 = 5
5 is less than 10, so write 5 in the ones column. There is nothing to carry.
Add the tens. Now the next column to the left.
6 + 8 = 14
14 tens is 1 hundred and 4 tens. Write 4 in the tens column and carry 1 to the hundreds column.
Add the hundreds. Now the next column to the left. Add the 1 we carried as well.
5 + 2 + 1 = 8 (the red digit is the carry)
8 is less than 10, so write 8 in the hundreds column. There is nothing to carry.
Add the thousands. Now the next column to the left.
4 + 7 = 11
11 thousands is 1 ten thousand and 1 thousand. Write 1 in the thousands column and carry 1 to the ten thousands column.
Add the ten thousands. Now the next column to the left. Add the 1 we carried as well.
8 + 4 + 1 = 13 (the red digit is the carry)
13 ten thousands is 1 hundred thousand and 3 ten thousands. Write 3 in the ten thousands column and carry 1 to the hundred thousands column.
Write the carried digit. There are no digits left in the hundred thousands column, but we carried 1 into it.
There is only one digit here, so write 1 in the hundred thousands column.
Read the answer. Check that the answer has a sensible number of digits, then write it.
HTh TTh Th H T O
8 4 5 6 0
+ 4 7 2 8 5
1 3 1 8 4 5
1 1 1

84,560 + 47,285 = 131,845

Check. Use the opposite (inverse) operation: 131,845 − 47,285 = 84,560. ✓
Estimate: 85,000 + 47,000 = 132,000 (rounding to the nearest 1,000). It is close to the exact answer. ✓

Watch out. Do not forget the carried digits. Write each one small under the answer line, and add it in the next column.

(b)How many more people live in Brigg than in Denby?

Method. Write the bigger number on top. Take away the ones first, then the tens, hundreds and so on. If a top digit is too small, exchange 1 from the next column.

Line up the digits in columns. The bigger number goes on top. Ones go under ones, tens under tens, hundreds under hundreds and so on. The numbers have different lengths, so line them up on the right.
HTh TTh Th H T O
1 2 6 4 0 0
− 9 1 7 5 0
Subtract the ones. Always start with the ones, on the right.
0 − 0 = 0
Exchange, then subtract the tens. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
0 tens is not enough to take away 5. We need to exchange.
Exchange 1 hundred for 10 tens: 4 hundreds become 3 hundreds, and 0 tens become 10 tens.
10 − 5 = 5
Exchange, then subtract the hundreds. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
3 hundreds is not enough to take away 7. We need to exchange.
(There were 4 hundreds to start with, but we gave one away in an earlier exchange.)
Exchange 1 thousand for 10 hundreds: 6 thousands become 5 thousands, and 3 hundreds become 13 hundreds.
13 − 7 = 6
Subtract the thousands. Now the next column to the left.
(This digit is 5 now, not 6, because we exchanged from this column.)
5 − 1 = 4
Exchange, then subtract the ten thousands. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
2 ten thousands is not enough to take away 9. We need to exchange.
Exchange 1 hundred thousand for 10 ten thousands: 1 hundred thousand become 0 hundred thousands, and 2 ten thousands become 12 ten thousands.
12 − 9 = 3
Bring down the hundred thousands digit. The bottom number has no digit in this column, so there is nothing to take away.
(This digit is 0 now, not 1, because we exchanged from this column.)
Nothing to take away in this column, so the digit stays: 0.
We do not write a 0 at the front of an answer, so leave this space empty.
Read the answer. Check that the answer has a sensible number of digits, then write it.
HTh TTh Th H T O
01 122 56 134 100 0
− 9 1 7 5 0
3 4 6 5 0

126,400 − 91,750 = 34,650

Check. Use the opposite (inverse) operation: 34,650 + 91,750 = 126,400. ✓
Estimate: 126,000 − 92,000 = 34,000 (rounding to the nearest 1,000). It is close to the exact answer. ✓

Watch out. The bottom digit is always taken from the top digit. Do not turn it round: 3 − 7 is not 4. Exchange instead.

(c)What is the total population of all four towns?

Method. Write the numbers in columns with the same place values lined up. Add the ones first, then the tens, hundreds and so on. If a column makes 10 or more, carry to the next column.

Line up the digits in columns. Ones go under ones, tens under tens, hundreds under hundreds and so on, so every digit is added to digits of the same value. The numbers have different lengths, so line them up on the right.
HTh TTh Th H T O
8 4 5 6 0
+ 1 2 6 4 0 0
+ 4 7 2 8 5
+ 9 1 7 5 0
Add the ones. Always start with the ones, on the right.
0 + 0 + 5 + 0 = 5
5 is less than 10, so write 5 in the ones column. There is nothing to carry.
Add the tens. Now the next column to the left.
6 + 0 + 8 + 5 = 19
19 tens is 1 hundred and 9 tens. Write 9 in the tens column and carry 1 to the hundreds column.
Add the hundreds. Now the next column to the left. Add the 1 we carried as well.
5 + 4 + 2 + 7 + 1 = 19 (the red digit is the carry)
19 hundreds is 1 thousand and 9 hundreds. Write 9 in the hundreds column and carry 1 to the thousands column.
Add the thousands. Now the next column to the left. Add the 1 we carried as well.
4 + 6 + 7 + 1 + 1 = 19 (the red digit is the carry)
19 thousands is 1 ten thousand and 9 thousands. Write 9 in the thousands column and carry 1 to the ten thousands column.
Add the ten thousands. Now the next column to the left. Add the 1 we carried as well.
8 + 2 + 4 + 9 + 1 = 24 (the red digit is the carry)
24 ten thousands is 2 hundred thousands and 4 ten thousands. Write 4 in the ten thousands column and carry 2 to the hundred thousands column.
Add the hundred thousands. Now the next column to the left. Add the 2 we carried as well.
1 + 2 = 3 (the red digit is the carry)
3 is less than 10, so write 3 in the hundred thousands column. There is nothing to carry.
Read the answer. Check that the answer has a sensible number of digits, then write it.
HTh TTh Th H T O
8 4 5 6 0
+ 1 2 6 4 0 0
+ 4 7 2 8 5
+ 9 1 7 5 0
3 4 9 9 9 5
2 1 1 1

84,560 + 126,400 + 47,285 + 91,750 = 349,995

Check. Use the opposite (inverse) operation: 349,995 − 126,400 − 47,285 − 91,750 = 84,560. ✓
Estimate: 85,000 + 126,000 + 47,000 + 92,000 = 350,000 (rounding to the nearest 1,000). It is close to the exact answer. ✓

Watch out. Do not forget the carried digits. Write each one small under the answer line, and add it in the next column.

Done

FThe bar chart shows the number of people at a festival each day. Use it to answer the questions.

(a)How many people went on Thursday?

(b)How many more people went on Tuesday than on Friday?

(c)How many people went on Monday and Tuesday together?

Done

PSUse the bar chart. Which two days had a total of 74,000 people?

Done

FComplete the bank statement. Write the balance after each line.

What happened	Money in	Money out	Balance
Opening balance			£2,450
Wages	£1,865		?
Rent		£950	?
Phone bill		£48	?
Shopping		£215	?
Gift	£150		?

Done

PSHere is part of a bank statement.

What happened	Money in	Money out
Wages	£3,200
Rent		£1,450
Car repair		£875
Holiday		£2,230
Refund	£640

How much more money went out than came in?

Done

FUse the bus timetable.

Bus stop	Time
School	08:15
Park	08:32
Library	08:47
Station	09:05
Market	09:38

How many minutes does the bus take for each journey?

(a)Park to Station

(b)Library to Market

(c)School to Market

Done

PSA plane flies a route in four stages.

Stage	Distance (km)
London to New York	5,570
New York to Los Angeles	3,940
Los Angeles to Tokyo	8,830
Tokyo to Delhi	5,840

What is the total distance of the whole route? How much of the route is not the middle two stages?

(a)The total distance of the whole route

Method. Write the numbers in columns with the same place values lined up. Add the ones first, then the tens, hundreds and so on. If a column makes 10 or more, carry to the next column.

Line up the digits in columns. Ones go under ones, tens under tens, hundreds under hundreds and so on, so every digit is added to digits of the same value.
TTh Th H T O
5 5 7 0
+ 3 9 4 0
+ 8 8 3 0
+ 5 8 4 0
Add the ones. Always start with the ones, on the right.
0 + 0 + 0 + 0 = 0
0 is less than 10, so write 0 in the ones column. There is nothing to carry.
Add the tens. Now the next column to the left.
7 + 4 + 3 + 4 = 18
18 tens is 1 hundred and 8 tens. Write 8 in the tens column and carry 1 to the hundreds column.
Add the hundreds. Now the next column to the left. Add the 1 we carried as well.
5 + 9 + 8 + 8 + 1 = 31 (the red digit is the carry)
31 hundreds is 3 thousands and 1 hundred. Write 1 in the hundreds column and carry 3 to the thousands column.
Add the thousands. Now the next column to the left. Add the 3 we carried as well.
5 + 3 + 8 + 5 + 3 = 24 (the red digit is the carry)
24 thousands is 2 ten thousands and 4 thousands. Write 4 in the thousands column and carry 2 to the ten thousands column.
Write the carried digit. There are no digits left in the ten thousands column, but we carried 2 into it.
There is only one digit here, so write 2 in the ten thousands column.
Read the answer. Check that the answer has a sensible number of digits, then write it.
TTh Th H T O
5 5 7 0
+ 3 9 4 0
+ 8 8 3 0
+ 5 8 4 0
2 4 1 8 0
2 3 1

5,570 + 3,940 + 8,830 + 5,840 = 24,180

Check. Use the opposite (inverse) operation: 24,180 − 3,940 − 8,830 − 5,840 = 5,570. ✓
Estimate: 5,600 + 3,900 + 8,800 + 5,800 = 24,100 (rounding to the nearest 100). It is close to the exact answer. ✓

Watch out. Do not forget the carried digits. Write each one small under the answer line, and add it in the next column.

(b)The distance of the first and last stages together

Done

PSA shop records its sales.

Day	Sales
Monday	£12,450
Tuesday	£9,875
Wednesday	£15,320
Thursday	£11,690

The shop’s costs for the four days were £27,500. How much money is left after the costs are paid?

Done

PSFor each calculation, choose the best method (mental or column) and work out the answer. Explain your choice.

(a)4,999 + 3,001

(b)45,368 + 27,594

Method. Write the numbers in columns with the same place values lined up. Add the ones first, then the tens, hundreds and so on. If a column makes 10 or more, carry to the next column.

Line up the digits in columns. Ones go under ones, tens under tens, hundreds under hundreds and so on, so every digit is added to digits of the same value.
TTh Th H T O
4 5 3 6 8
+ 2 7 5 9 4
Add the ones. Always start with the ones, on the right.
8 + 4 = 12
12 ones is 1 ten and 2 ones. Write 2 in the ones column and carry 1 to the tens column.
Add the tens. Now the next column to the left. Add the 1 we carried as well.
6 + 9 + 1 = 16 (the red digit is the carry)
16 tens is 1 hundred and 6 tens. Write 6 in the tens column and carry 1 to the hundreds column.
Add the hundreds. Now the next column to the left. Add the 1 we carried as well.
3 + 5 + 1 = 9 (the red digit is the carry)
9 is less than 10, so write 9 in the hundreds column. There is nothing to carry.
Add the thousands. Now the next column to the left.
5 + 7 = 12
12 thousands is 1 ten thousand and 2 thousands. Write 2 in the thousands column and carry 1 to the ten thousands column.
Add the ten thousands. Now the next column to the left. Add the 1 we carried as well.
4 + 2 + 1 = 7 (the red digit is the carry)
7 is less than 10, so write 7 in the ten thousands column. There is nothing to carry.
Read the answer. Check that the answer has a sensible number of digits, then write it.
TTh Th H T O
4 5 3 6 8
+ 2 7 5 9 4
7 2 9 6 2
1 1 1

45,368 + 27,594 = 72,962

Check. Use the opposite (inverse) operation: 72,962 − 27,594 = 45,368. ✓
Estimate: 45,000 + 28,000 = 73,000 (rounding to the nearest 1,000). It is close to the exact answer. ✓

Watch out. Do not forget the carried digits. Write each one small under the answer line, and add it in the next column.

(d)306,040 − 148,975

Method. Write the bigger number on top. Take away the ones first, then the tens, hundreds and so on. If a top digit is too small, exchange 1 from the next column.

Line up the digits in columns. The bigger number goes on top. Ones go under ones, tens under tens, hundreds under hundreds and so on.
HTh TTh Th H T O
3 0 6 0 4 0
− 1 4 8 9 7 5
Exchange, then subtract the ones. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
0 ones is not enough to take away 5. We need to exchange.
Exchange 1 ten for 10 ones: 4 tens become 3 tens, and 0 ones become 10 ones.
10 − 5 = 5
Exchange, then subtract the tens. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
3 tens is not enough to take away 7. We need to exchange.
(There were 4 tens to start with, but we gave one away in an earlier exchange.)
The hundreds has 0, so there is nothing to exchange from there. We start further to the left, at the thousands.
Exchange 1 thousand for 10 hundreds: 6 thousands become 5 thousands, and 0 hundreds become 10 hundreds.
Exchange 1 hundred for 10 tens: 10 hundreds become 9 hundreds, and 3 tens become 13 tens.
13 − 7 = 6
Subtract the hundreds. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 9 = 0
Exchange, then subtract the thousands. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
5 thousands is not enough to take away 8. We need to exchange.
(There were 6 thousands to start with, but we gave one away in an earlier exchange.)
The ten thousands has 0, so there is nothing to exchange from there. We start further to the left, at the hundred thousands.
Exchange 1 hundred thousand for 10 ten thousands: 3 hundred thousands become 2 hundred thousands, and 0 ten thousands become 10 ten thousands.
Exchange 1 ten thousand for 10 thousands: 10 ten thousands become 9 ten thousands, and 5 thousands become 15 thousands.
15 − 8 = 7
Subtract the ten thousands. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 4 = 5
Subtract the hundred thousands. Now the next column to the left.
(This digit is 2 now, not 3, because we exchanged from this column.)
2 − 1 = 1
Read the answer. Check that the answer has a sensible number of digits, then write it.
HTh TTh Th H T O
23 90 156 90 134 100
− 1 4 8 9 7 5
1 5 7 0 6 5

306,040 − 148,975 = 157,065

Check. Use the opposite (inverse) operation: 157,065 + 148,975 = 306,040. ✓
Estimate: 310,000 − 150,000 = 160,000 (rounding to the nearest 10,000). It is close to the exact answer. ✓

Watch out. You cannot exchange from a 0. Look left for the first column that is not 0, then pass the exchange along one column at a time. Each 0 in the middle becomes 9.

Done

RLeo looks at the bar chart and says, “The Thursday bar is 45,000 because it is between 40,000 and 50,000.” What mistake did Leo make? What is the correct number?

Done

RZainab says, “Ashby and Calder together have more people than Brigg.” Is she right? Use an estimate first, then check with the exact numbers.

Town	Population
Ashby	84,560
Brigg	126,400
Calder	47,285
Denby	91,750

Done

RWhich one is the odd one out? Explain how you know.
A 20,000 + 12,500 B 50,000 − 17,500 C 16,250 + 16,250 D 40,000 − 7,000

Done

RThe table shows how many tickets were sold at a fair. Some numbers are missing. Find all the missing numbers.

	Adults	Children	Total
Saturday	12,450	?	20,100
Sunday	?	9,850	18,300
Total	?	?	?

Done

Chapter summary

Skill	Method
Column addition	Line up ones under ones (on the right). Add from the right. If a column makes 10 or more, write the ones digit and carry. With three numbers the carry can be 2.
Column subtraction	Bigger number on top. Subtract from the right. If the top digit is smaller, exchange 1 from the next column (10 more in this column).
Zeros in the top number	You cannot exchange from a 0. Go left to the first column that is not 0. Each 0 you pass becomes 9. 1,000,000 − 468,325 works this way.
Mental strategies	Partition; bridge through 10, 100 or 1,000; compensate (add 9,999 = add 10,000, take away 1); number facts (75 + 25 = 100, so 7,500 + 2,500 = 10,000); count up; near doubles.
Estimating	Round each number (to the nearest 10, 100, 1,000, 10,000 or 100,000), then do the easy calculation. Use ≈. Round to a smaller place value for a closer estimate.
Level of accuracy	Choose how much to round to fit the question. If the estimate is close to a limit, use a closer estimate or the exact answer.
Checking	Use the inverse: check an addition by subtracting and a subtraction by adding. Compare the answer with your estimate. A wrong ones digit means a wrong answer.
Missing digits	Work from the ones column. Use the carry. Turn subtractions into additions with the inverse. The answer is always unique.
Bar models	Part-whole: parts added together make the whole. Comparison: the gap is the difference. Draw the bars before you choose the operation.
Multi-step problems	Read, draw a bar model, plan the steps, calculate, check that you answered the question. Do not stop after step 1.
Tables, charts, timetables, bank statements	Read the scale carefully. Count on to the next o’clock for time gaps. Money in adds and money out subtracts.

Can you…?

I can add numbers with up to 6 digits using the column method, including three numbers and carrying.
I can subtract numbers with up to 6 digits using the column method, with exchanging across zeros.
I can add and subtract in my head by partitioning, bridging, compensating, using number facts and near doubles.
I can round to estimate an answer and choose how accurate I need to be.
I can check an answer with the inverse operation.
I can find missing digits and missing numbers.
I can use part-whole and comparison bar models.
I can solve multi-step problems and explain which operations and methods I chose.
I can solve problems from tables, bar charts, timetables and bank statements.

Key terms

Column method: A written way to add or subtract. Numbers go one under the other with the same place values lined up.
Carry: When a column adds to 10 or more, you write the ones digit and move the rest to the next column.
Exchange: In subtraction, swapping 1 of the next column for 10 of this column (for example 1 thousand for 10 hundreds). Also called borrowing.
Sum / total: The answer to an addition.
Difference: How far apart two numbers are. It is the answer to a subtraction.
Inverse: The opposite operation. Addition and subtraction undo each other.
Estimate: A close answer found by rounding the numbers first.
Accuracy: How close an answer is to the exact answer.
Partition: To split a number into parts, such as 36,400 = 30,000 + 6,000 + 400.
Bridging: Jumping to the next round number (10, 100 or 1,000) and then doing the rest.
Compensate: To use an easier number and then fix the answer, such as adding 10,000 and taking away 1 to add 9,999.
Number bond: A pair of numbers that make a given total, such as 36 + 64 = 100.
Near double: A sum of two numbers that are almost the same.
Bar model: A drawing of bars that shows the numbers in a problem. Part-whole models show parts and a whole. Comparison models show two amounts side by side.
Balance: The amount of money in a bank account.
Multi-step problem: A problem that needs more than one calculation to solve.

Total 19 marks. Allow about 19 minutes.

Exercise 2X SATs-style questions

Non-calculator

FWork out 456,208 + 235,794.[1 mark]

Method. Write the numbers in columns with the same place values lined up. Add the ones first, then the tens, hundreds and so on. If a column makes 10 or more, carry to the next column.

Line up the digits in columns. Ones go under ones, tens under tens, hundreds under hundreds and so on, so every digit is added to digits of the same value.
HTh TTh Th H T O
4 5 6 2 0 8
+ 2 3 5 7 9 4
Add the ones. Always start with the ones, on the right.
8 + 4 = 12
12 ones is 1 ten and 2 ones. Write 2 in the ones column and carry 1 to the tens column.
Add the tens. Now the next column to the left. Add the 1 we carried as well.
0 + 9 + 1 = 10 (the red digit is the carry)
10 tens is 1 hundred and 0 tens. Write 0 in the tens column and carry 1 to the hundreds column.
Add the hundreds. Now the next column to the left. Add the 1 we carried as well.
2 + 7 + 1 = 10 (the red digit is the carry)
10 hundreds is 1 thousand and 0 hundreds. Write 0 in the hundreds column and carry 1 to the thousands column.
Add the thousands. Now the next column to the left. Add the 1 we carried as well.
6 + 5 + 1 = 12 (the red digit is the carry)
12 thousands is 1 ten thousand and 2 thousands. Write 2 in the thousands column and carry 1 to the ten thousands column.
Add the ten thousands. Now the next column to the left. Add the 1 we carried as well.
5 + 3 + 1 = 9 (the red digit is the carry)
9 is less than 10, so write 9 in the ten thousands column. There is nothing to carry.
Add the hundred thousands. Now the next column to the left.
4 + 2 = 6
6 is less than 10, so write 6 in the hundred thousands column. There is nothing to carry.
Read the answer. Check that the answer has a sensible number of digits, then write it.
HTh TTh Th H T O
4 5 6 2 0 8
+ 2 3 5 7 9 4
6 9 2 0 0 2
1 1 1 1

456,208 + 235,794 = 692,002

Check. Use the opposite (inverse) operation: 692,002 − 235,794 = 456,208. ✓
Estimate: 460,000 + 240,000 = 700,000 (rounding to the nearest 10,000). It is close to the exact answer. ✓

Watch out. Do not forget the carried digits. Write each one small under the answer line, and add it in the next column.

Done

FWork out 800,000 − 354,278.[1 mark]

Method. Write the bigger number on top. Take away the ones first, then the tens, hundreds and so on. If a top digit is too small, exchange 1 from the next column.

Line up the digits in columns. The bigger number goes on top. Ones go under ones, tens under tens, hundreds under hundreds and so on.
HTh TTh Th H T O
8 0 0 0 0 0
− 3 5 4 2 7 8
Exchange, then subtract the ones. The top digit is smaller than the bottom digit, so we cannot take it away yet. We swap 1 from the next column.
0 ones is not enough to take away 8. We need to exchange.
The tens, hundreds, thousands and ten thousands have 0, so there is nothing to exchange from there. We start further to the left, at the hundred thousands.
Exchange 1 hundred thousand for 10 ten thousands: 8 hundred thousands become 7 hundred thousands, and 0 ten thousands become 10 ten thousands.
Exchange 1 ten thousand for 10 thousands: 10 ten thousands become 9 ten thousands, and 0 thousands become 10 thousands.
Exchange 1 thousand for 10 hundreds: 10 thousands become 9 thousands, and 0 hundreds become 10 hundreds.
Exchange 1 hundred for 10 tens: 10 hundreds become 9 hundreds, and 0 tens become 10 tens.
Exchange 1 ten for 10 ones: 10 tens become 9 tens, and 0 ones become 10 ones.
10 − 8 = 2
Subtract the tens. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 7 = 2
Subtract the hundreds. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 2 = 7
Subtract the thousands. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 4 = 5
Subtract the ten thousands. Now the next column to the left.
(This digit is 9 now, not 0, because we exchanged from this column.)
9 − 5 = 4
Subtract the hundred thousands. Now the next column to the left.
(This digit is 7 now, not 8, because we exchanged from this column.)
7 − 3 = 4
Read the answer. Check that the answer has a sensible number of digits, then write it.
HTh TTh Th H T O
78 90 90 90 90 100
− 3 5 4 2 7 8
4 4 5 7 2 2

800,000 − 354,278 = 445,722

Check. Use the opposite (inverse) operation: 445,722 + 354,278 = 800,000. ✓
Estimate: 800,000 − 350,000 = 450,000 (rounding to the nearest 10,000). It is close to the exact answer. ✓

Watch out. You cannot exchange from a 0. Look left for the first column that is not 0, then pass the exchange along one column at a time. Each 0 in the middle becomes 9.

Done

FWhich is the best estimate for 48,920 + 31,070? Tick one box. A 8,000 B 80,000 C 800,000 D 18,000[1 mark]

Done

FWrite the missing number. ? − 24,500 = 61,250[1 mark]

Done

FA school wants to collect 65,000 bottle tops. In week 1 it collects 28,450 and in week 2 it collects 19,375. How many more bottle tops does it need? Show your working.[2 marks]

Done

RAmira says that 48,360 − 29,875 = 28,485. Use an estimate to show that Amira’s answer is wrong. Then write the correct answer.[2 marks]

Done

RSam says that 100,000 − 37,625 = 62,475. Use addition to show that Sam is wrong. Then write the correct answer.[2 marks]

Done

PSThe table shows how many people visit a museum each month.

Month	Visitors
June	45,680
July	63,250
August	71,940

[3 marks]

(a)How many more people visit the museum in July than in June? (1 mark)

(b)How many people visit the museum in the three months altogether? (2 marks)

Done

RDev thinks of a number. He subtracts 12,500. Then he adds 30,000. His answer is 100,000. What number did Dev think of first? Show how you know.[3 marks]

Done

PSNoah has 38,250 points. Priya has 12,500 more points than Noah. Noah, Priya and Leo have 150,000 points altogether. How many points does Leo have? Show your working.[3 marks]

Done

← Chapter 1: Place value to 1,000,000 Chapter 3: Multiples, factors, primes, squares and cubes →

Year 5 Column Addition and Subtraction: worked examples and questions with answers

By the end of this chapter you will be able to

You should already know

Where this leads

Exercise P Do you remember?

Column addition with bigger numbers

Key ideas

Watch out

Exercise 2A Column addition

Column subtraction with bigger numbers

Key ideas

Watch out

Exercise 2B Column subtraction

Adding and subtracting in your head

Key ideas

Watch out

Exercise 2C Mental strategies

Estimating, rounding and checking

Key ideas

Watch out

Exercise 2D Estimating, inverse operations and missing numbers

Bar models and multi-step problems

Key ideas

Watch out

Exercise 2E Bar models and multi-step problems

Missing digits and number puzzles

Key ideas

Watch out

Exercise 2F Missing digits and puzzles

Tables, charts, timetables and bank statements

Key ideas

Watch out

Exercise 2G Tables, charts, timetables and bank statements

Chapter summary

Can you…?

Key terms

Exercise 2Z Check-up (non-calculator)

Exercise 2X SATs-style questions