Year 5: Bar models and multi-step problems
Part of the Year 5 column addition and subtraction chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Bar models and multi-step problems
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.
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.
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 |
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?
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?
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 4 59 09 010 00 − 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?
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 4 511 212 314 510 0− 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.
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.
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 4 59 09 09 010 0− 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.
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.
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 2 39 09 010 00 − 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.
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 5 614 513 410 00 − 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.
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.
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 4 510 07 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.Exercise 2E Bar models and multi-step problems
Non-calculatorKey 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 4 5 2 0 0 + 3 8 7 5 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.0 + 5 = 5
5 is less than 10, so write 5 in the tens column. There is nothing to carry. - Add the hundreds. Now the next column to the left.2 + 7 = 9
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 + 8 = 13
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.4 + 3 + 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 4 5 2 0 0 + 3 8 7 5 0 8 3 9 5 0 1
45,200 + 38,750 = 83,950
Check. Use the opposite (inverse) operation: 83,950 − 38,750 = 45,200. ✓
Estimate: 45,000 + 39,000 = 84,000 (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.
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. The numbers have different lengths, so line them up on the right.
HTh TTh Th H T O 1 2 6 4 0 0 + 8 7 9 5 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.0 + 5 = 5
5 is less than 10, so write 5 in the tens column. There is nothing to carry. - Add the hundreds. Now the next column to the left.4 + 9 = 13
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.6 + 7 + 1 = 14 (the red digit is the carry)
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.2 + 8 + 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.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 1 2 6 4 0 0 + 8 7 9 5 0 2 1 4 3 5 0 1 1 1
126,400 + 87,950 = 214,350
Check. Use the opposite (inverse) operation: 214,350 − 87,950 = 126,400. ✓
Estimate: 126,000 + 88,000 = 214,000 (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.
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.
HTh TTh Th H T O 3 5 0 0 0 0 + 2 4 9 5 0 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.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 + 5 = 5
5 is less than 10, so write 5 in the hundreds column. There is nothing to carry. - Add the thousands. Now the next column to the left.0 + 9 = 9
9 is less than 10, so write 9 in the thousands column. There is nothing to carry. - Add the ten thousands. Now the next column to the left.5 + 4 = 9
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.3 + 2 = 5
5 is less than 10, so write 5 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 5 0 0 0 0 + 2 4 9 5 0 0 5 9 9 5 0 0
350,000 + 249,500 = 599,500
Check. Use the opposite (inverse) operation: 599,500 − 249,500 = 350,000. ✓
Estimate: 350,000 + 250,000 = 600,000 (rounding to the nearest 10,000). It is close to the exact answer. ✓
Watch out. Start on the right with the ones, not on the left.
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.
TTh Th H T O 8 0 0 0 0 − 3 4 5 0 0 - Subtract the ones. Always start with the ones, on the right.0 − 0 = 0
- Subtract the tens. Now the next column to the left.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.0 hundreds is not enough to take away 5. We need to exchange.
The thousands has 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.
10 − 5 = 5 - 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 − 3 = 4 - Read the answer. Check that the answer has a sensible number of digits, then write it.
TTh Th H T O 7 89 010 00 0 − 3 4 5 0 0 4 5 5 0 0
80,000 − 34,500 = 45,500
Check. Use the opposite (inverse) operation: 45,500 + 34,500 = 80,000. ✓
Estimate: 80,000 − 35,000 = 45,000 (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.
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.
HTh TTh Th H T O 2 5 0 0 0 0 − 1 6 8 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.
The 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: 5 ten thousands become 4 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.
10 − 5 = 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 9 now, not 0, because we exchanged from this column.)
9 − 8 = 1 - 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.4 ten thousands is not enough to take away 6. We need to exchange.
(There were 5 ten thousands to start with, but we gave one away in an earlier exchange.)
Exchange 1 hundred thousand for 10 ten thousands: 2 hundred thousands become 1 hundred thousand, and 4 ten thousands become 14 ten thousands.
14 − 6 = 8 - Subtract the hundred thousands. Now the next column to the left.(This digit is 1 now, not 2, because we exchanged from this column.)
1 − 1 = 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 1 214 59 09 010 00 − 1 6 8 7 5 0 8 1 2 5 0
250,000 − 168,750 = 81,250
Check. Use the opposite (inverse) operation: 81,250 + 168,750 = 250,000. ✓
Estimate: 250,000 − 170,000 = 80,000 (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.
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.
HTh TTh Th H T O 9 0 0 0 0 0 − 4 5 7 3 0 0 - Subtract the ones. Always start with the ones, on the right.0 − 0 = 0
- Subtract the tens. Now the next column to the left.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.0 hundreds is not enough to take away 3. We need to exchange.
The 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: 9 hundred thousands become 8 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.
10 − 3 = 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 − 7 = 2 - 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 8 now, not 9, because we exchanged from this column.)
8 − 4 = 4 - Read the answer. Check that the answer has a sensible number of digits, then write it.
HTh TTh Th H T O 8 99 09 010 00 0 − 4 5 7 3 0 0 4 4 2 7 0 0
900,000 − 457,300 = 442,700
Check. Use the opposite (inverse) operation: 442,700 + 457,300 = 900,000. ✓
Estimate: 900,000 − 460,000 = 440,000 (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.
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.
TTh Th H T O 7 2 5 0 0 − 4 8 2 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: 5 hundreds become 4 hundreds, and 0 tens become 10 tens.
10 − 5 = 5 - Subtract the hundreds. Now the next column to the left.(This digit is 4 now, not 5, because we exchanged from this column.)
4 − 2 = 2 - 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.2 thousands is not enough to take away 8. We need to exchange.
Exchange 1 ten thousand for 10 thousands: 7 ten thousands become 6 ten thousands, and 2 thousands become 12 thousands.
12 − 8 = 4 - Subtract the ten thousands. Now the next column to the left.(This digit is 6 now, not 7, because we exchanged from this column.)
6 − 4 = 2 - Read the answer. Check that the answer has a sensible number of digits, then write it.
TTh Th H T O 6 712 24 510 00 − 4 8 2 5 0 2 4 2 5 0
72,500 − 48,250 = 24,250
Check. Use the opposite (inverse) operation: 24,250 + 48,250 = 72,500. ✓
Estimate: 73,000 − 48,000 = 25,000 (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.
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.
HTh TTh Th H T O 5 0 0 0 0 0 − 2 7 6 4 0 0 - Subtract the ones. Always start with the ones, on the right.0 − 0 = 0
- Subtract the tens. Now the next column to the left.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.0 hundreds is not enough to take away 4. We need to exchange.
The 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.
10 − 4 = 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 − 6 = 3 - 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 − 7 = 2 - 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 - Read the answer. Check that the answer has a sensible number of digits, then write it.
HTh TTh Th H T O 4 59 09 010 00 0 − 2 7 6 4 0 0 2 2 3 6 0 0
500,000 − 276,400 = 223,600
Check. Use the opposite (inverse) operation: 223,600 + 276,400 = 500,000. ✓
Estimate: 500,000 − 280,000 = 220,000 (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.
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. 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 − 6 4 5 0 0 0 - Subtract the ones. Always start with the ones, on the right.0 − 0 = 0
- Subtract the tens. Now the next column to the left.0 − 0 = 0
- Subtract the hundreds. Now the next column to the left.0 − 0 = 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.0 thousands is not enough to take away 5. We need to exchange.
The 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.
10 − 5 = 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 − 4 = 5 - 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 − 6 = 3 - 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 0 19 09 010 00 0 0 − 6 4 5 0 0 0 3 5 5 0 0 0
1,000,000 − 645,000 = 355,000
Check. Use the opposite (inverse) operation: 355,000 + 645,000 = 1,000,000. ✓
Estimate: 1,000,000 − 650,000 = 350,000 (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.
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. Bigger has 8,750 more than Smaller, so we add.24,500 + 8,750
- Work it out. Use the column method because the numbers are big.24,500 + 8,750 = 33,250
TTh Th H T O 2 4 5 0 0 + 8 7 5 0 3 3 2 5 0 1 1 - Write the answer. 33,250
Check. Use the opposite operation: 33,250 − 8,750 = 24,500. ✓
Watch out. “More than” does not always mean add. Draw the bars first, then think about which bar you are finding.
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. Smaller has 37,500 fewer than Bigger, so we take away.91,000 − 37,500
- Work it out. Use the column method because the numbers are big.91,000 − 37,500 = 53,500
TTh Th H T O 8 910 110 00 0 − 3 7 5 0 0 5 3 5 0 0 - Write the answer. 53,500
Check. Use the opposite operation: 53,500 + 37,500 = 91,000. ✓
Watch out. “More than” does not always mean add. Draw the bars first, then think about which bar you are finding.
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. Bigger has 99,950 more than Smaller, so we add.300,050 + 99,950
- Work it out. Use the column method because the numbers are big.300,050 + 99,950 = 400,000
HTh TTh Th H T O 3 0 0 0 5 0 + 9 9 9 5 0 4 0 0 0 0 0 1 1 1 1 - Write the answer. 400,000
Check. Use the opposite operation: 400,000 − 99,950 = 300,050. ✓
Watch out. “More than” does not always mean add. Draw the bars first, then think about which bar you are finding.
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.8,849 m − 1,345 m
- Work it out. Use the column method because the numbers are big.8,849 m − 1,345 m = 7,504 m
Th H T O 8 8 4 9 − 1 3 4 5 7 5 0 4 - Write the answer. 7,504 m
Check. Use the opposite operation: 1,345 m + 7,504 m = 8,849 m. ✓
Watch out. “More than” does not always mean add. Draw the bars first, then think about which bar you are finding.
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 75,000. It is split into the two amounts used and the part that is left.
- Step 1: add the two amounts. Put the two parts together.18,250 + 24,780 = 43,030
TTh Th H T O 1 8 2 5 0 + 2 4 7 8 0 4 3 0 3 0 1 1 1 - Step 2: take the total away from the start. What is left is the whole minus the parts that were used.75,000 − 43,030 = 31,970
TTh Th H T O 7 4 59 010 00 − 4 3 0 3 0 3 1 9 7 0 - Write the answer. 31,970 litres
Check. Add everything back: 18,250 + 24,780 + 31,970 = 75,000, 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.
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, 36,850.
- Step 1: find the amount for Marsh Bay. Marsh Bay has 36,850 less, so we subtract.125,400 − 36,850 = 88,550
HTh TTh Th H T O 0 111 214 513 410 00 − 3 6 8 5 0 8 8 5 5 0 - Step 2: find the total. Add the two amounts: Hull and Marsh Bay.125,400 + 88,550 = 213,950
HTh TTh Th H T O 1 2 5 4 0 0 + 8 8 5 5 0 2 1 3 9 5 0 1 1 - Write the answer. 213,950 people
Check. The difference between the two amounts should be 36,850: 125,400 − 88,550 = 36,850. ✓
Watch out. A common mistake is to stop after step 1. The question asks for the total of BOTH amounts.
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. 15,750 get off, so we subtract.48,600 − 15,750 = 32,850
TTh Th H T O 4 7 815 610 00 − 1 5 7 5 0 3 2 8 5 0 - Step 2: add. Now 22,400 get on, so we add to the new total.32,850 + 22,400 = 55,250
TTh Th H T O 3 2 8 5 0 + 2 2 4 0 0 5 5 2 5 0 1 - Write the answer. 55,250 people
Check. Work backwards: 55,250 − 22,400 = 32,850 and 32,850 + 15,750 = 48,600. ✓
Watch out. Follow the story in order. The first thing that happens is the first calculation.
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.18,650 km + 27,340 km = 45,990 km
TTh Th H T O 1 8 6 5 0 + 2 7 3 4 0 4 5 9 9 0 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.52,000 km − 45,990 km = 6,010 km
TTh Th H T O 4 511 29 010 00 − 4 5 9 9 0 6 0 1 0 - Write the answer. 6,010 km
Check. Add the difference to the smaller amount: 45,990 km + 6,010 km = 52,000 km. ✓
Watch out. Take the SMALLER amount from the BIGGER amount to find the difference.
Key words. Change is the money you get back when you pay with more than the cost.
Method. This is a two-step problem. First add up how much everything costs. Then take the cost away from the money we have. What is left is the answer.
- Draw a bar model. The whole bar is the money we start with (or pay). It is split into the cost of each thing and what is left.
- Step 1: find the total cost. Add the prices.£12,495 + £3,750 + £8,920 = £25,165
TTh Th H T O 1 2 4 9 5 + 3 7 5 0 + 8 9 2 0 2 5 1 6 5 1 2 1 - Step 2: find the money left. Take the total cost away from the money we started with.£30,000 − £25,165 = £4,835
TTh Th H T O 2 39 09 09 010 0− 2 5 1 6 5 4 8 3 5 - Write the answer. £4,835 left
Check. Add the cost and what is left: £25,165 + £4,835 = £30,000, which is the money we started with. ✓
Watch out. Do not take away just one price. Everything was bought, so take away the total cost.
Method. This is a three-step problem. Work out each unknown amount in order, then add all three at the end.
- Draw the bars. Zainab is longer than Leo. Priya is longer than Zainab.
- Step 1: find how many Zainab has. Zainab has 18,400 more than Leo, so we add.45,600 + 18,400 = 64,000
TTh Th H T O 4 5 6 0 0 + 1 8 4 0 0 6 4 0 0 0 1 1 - Step 2: find how many Priya has. Priya has 9,250 more than Zainab, so we add.64,000 + 9,250 = 73,250
TTh Th H T O 6 4 0 0 0 + 9 2 5 0 7 3 2 5 0 1 - Step 3: add all three amounts. The question asks for the total of all three.45,600 + 64,000 + 73,250 = 182,850
HTh TTh Th H T O 4 5 6 0 0 + 6 4 0 0 0 + 7 3 2 5 0 1 8 2 8 5 0 1 1 - Write the answer. 182,850 points
Check. Check each comparison: 64,000 − 45,600 = 18,400 ✓ and 9,250 is the difference between Zainab and Priya. ✓
Watch out. Do not stop early. Each step gives a number you need for the next one, and the last step answers the question.
Method. Decide if the question is about a whole and its parts, or about comparing two separate amounts.
- Read the question. Mia and Noah each have their own stickers. We are not putting them together. We are comparing them.Question: “How many more does Mia have than Noah?”
- Choose the bar model. Model A has two bars side by side, with the difference marked. Model B puts the two amounts together into a total, which is not what the question asks.Model A matches.
- Answer the question. The difference is the bigger amount minus the smaller amount.24,500 − 8,750 = 15,750
TTh Th H T O 1 213 414 510 00 − 8 7 5 0 1 5 7 5 0
Check. 8,750 + 15,750 = 24,500. ✓
Watch out. Model B would be right for “How many stickers do Mia and Noah have altogether?”
A 24,500 + 8,750 + 6,425
B 24,500 − 8,750 − 6,425
C 24,500 − 8,750 + 6,425
D 24,500 − (8,750 + 6,425)
Method. Think about the story. Rolls are sold twice, so both amounts must be taken away. Then work out each choice to see which two give the same answer.
- Think about the story. The baker starts with 24,500. She sells 8,750 and then 6,425. Both are taken away.24,500 − 8,750 − 6,425
- Work out each choice. Do the brackets first.A: 24,500 + 8,750 + 6,425 = 39,675
B: 24,500 − 8,750 − 6,425 = 9,325
C: 24,500 − 8,750 + 6,425 = 22,175
D: 24,500 − (8,750 + 6,425) = 9,325 - Choose the two that match the story. B takes the two amounts away one after the other. D adds the two sold amounts first and then takes the total away. Both give 9,325.B and D
Check. Add everything back: 8,750 + 6,425 + 9,325 = 24,500. ✓
Watch out. Choice C adds the 6,425 on by mistake. Choice A adds both amounts to the start. Neither makes sense for rolls that are sold.
Method. Work out the problem step by step and see what 47,250 is.
- Do step 1. Ben scores 12,750 more than Aisha, so add.34,500 + 12,750 = 47,250
TTh Th H T O 3 4 5 0 0 + 1 2 7 5 0 4 7 2 5 0 1 - Compare with Ravi’s answer. 47,250 is Ben’s score, not the total.Ravi stopped after step 1.
- Do step 2. The question asks for both scores together.34,500 + 47,250 = 81,750
TTh Th H T O 3 4 5 0 0 + 4 7 2 5 0 8 1 7 5 0 1 - Write the correct answer. 81,750 points
Check. Aisha scores 34,500 and Ben scores 47,250. 47,250 − 34,500 = 12,750. ✓
Watch out. Read the last sentence of the question again before you write your answer. It tells you what to find.
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 the school had at the start. The parts are the laptop, the printer and the money left.
- Choose the operation. We know all the parts and want the whole, so we add.£8,450 + £2,375 + £12,600
- Work it out. Use the column method because the numbers are big.£8,450 + £2,375 + £12,600 = £23,425
TTh Th H T O 8 4 5 0 + 2 3 7 5 + 1 2 6 0 0 2 3 4 2 5 1 1 1 - Write the answer. £23,425
Check. Take the parts away again: £23,425 − £8,450 − £2,375 = £12,600. ✓
Watch out. This is a “working backwards” problem. The whole is bigger than every part, so you add, not subtract.
Method. Look at the comparison bar model. The difference is the gap between the two bars, so adding it to the smaller amount gives the bigger amount.
- Use Priya’s method. The missing number is the gap that is added to 417,500 to reach 600,000.417,500 + ? = 600,000, so ? = 600,000 − 417,500
- Work out the gap. Subtract in columns.600,000 − 417,500 = 182,500
HTh TTh Th H T O 5 69 09 010 00 0 − 4 1 7 5 0 0 1 8 2 5 0 0 - Check the other way. Add the gap to the smaller amount.417,500 + 182,500 = 600,000
HTh TTh Th H T O 4 1 7 5 0 0 + 1 8 2 5 0 0 6 0 0 0 0 0 1 1 1 - Answer. Yes, Priya is right. Both ways give 182,500.
Check. The gap added to the smaller amount gives the bigger amount: 417,500 + 182,500 = 600,000. ✓
Watch out. Finding the difference by counting up and by subtracting are both correct. They always give the same answer.