Year 6 Arithmetic (SATs) · Place value and the four operations · Lesson 1.4
Year 6 Arithmetic (SATs): Adding and subtracting large numbers
Part of the Year 6 Arithmetic (SATs) place value and the four operations chapter. Chapter overview · whole chapter on one page
1.4
Adding and subtracting large numbers
Story. In one year a city museum has 2,345,678 visitors. The next year it has 1,987,654. The manager adds the two totals in a column, carrying where needed.
Rule
Line the numbers up so that each digit is in its own column. Start at the ones column on the right.
In addition, if a column makes ten or more, write the ones digit and carry the tens digit to the next column.
In subtraction, if the top digit is smaller, borrow 10 from the column on its left.
Mental methods can split a number: 2,400,000 + 300,000 is 2,700,000 because 24 + 3 = 27 hundred thousand.
Key ideas
- Line up the place values before you start.
- Carry in addition when a column makes 10 or more.
- Borrow in subtraction when the top digit is too small.
- Check a subtraction by adding the answer back to the smaller number.
Worked example 1.4.1
Work out 4,718,260 + 2,396,845.
Method. Add column by column from the right, carrying where needed.
- Add the ones column. Write the ones digit of the total in this column, and carry the tens digit to the next column.0 + 5 + 0 = 5, write 5
- Add the tens column. Write the ones digit of the total in this column, and carry the tens digit to the next column.6 + 4 + 0 = 10, write 0, carry 1
- Add the hundreds column. Write the ones digit of the total in this column, and carry the tens digit to the next column.2 + 8 + 1 = 11, write 1, carry 1
- Add the thousands column. Write the ones digit of the total in this column, and carry the tens digit to the next column.8 + 6 + 1 = 15, write 5, carry 1
- Add the ten thousands column. Write the ones digit of the total in this column, and carry the tens digit to the next column.1 + 9 + 1 = 11, write 1, carry 1
- Add the hundred thousands column. Write the ones digit of the total in this column, and carry the tens digit to the next column.7 + 3 + 1 = 11, write 1, carry 1
- Add the millions column. Write the ones digit of the total in this column, and carry the tens digit to the next column.4 + 2 + 1 = 7, write 7
- Write the answer. 7,115,105
Check. Check the other way round: 2,396,845 + 4,718,260 = 7,115,105. ✓
Watch out. Do not forget to carry from one column to the next.
Worked example 1.4.2
Work out 6,004,517 - 2,398,649.
Method. Subtract column by column from the right, borrowing where needed.
- Subtract the ones column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.7 - 0 = 7, which is less than 9, so borrow 10: 17 - 9 = 8
- Subtract the tens column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.1 - 1 = 0, which is less than 4, so borrow 10: 10 - 4 = 6
- Subtract the hundreds column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.5 - 1 = 4, which is less than 6, so borrow 10: 14 - 6 = 8
- Subtract the thousands column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.4 - 1 = 3, which is less than 8, so borrow 10: 13 - 8 = 5
- Subtract the ten thousands column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.0 - 1 = -1, which is less than 9, so borrow 10: 9 - 9 = 0
- Subtract the hundred thousands column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.0 - 1 = -1, which is less than 3, so borrow 10: 9 - 3 = 6
- Subtract the millions column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.5 - 2 = 3
- Write the answer. 3,605,868
Check. Check: 3,605,868 + 2,398,649 = 6,004,517. ✓
Watch out. Borrowing across several zeros is the hardest case: check each column.
Worked example 1.4.3
Work out mentally: 2,400,000 + 300,000.
Method. Add the hundred thousands, not the individual digits.
- Split the numbers. 2,400,000 is 24 hundred thousand and 300,000 is 3 hundred thousand.24 + 3 = 27 hundred thousand
- Put it back together. Twenty-seven hundred thousand is 2,700,000.2,700,000
- Write the answer. 2,700,000
Check. 2,700,000 - 300,000 = 2,400,000. ✓
Exercise 1D Practice: adding and subtracting large numbers
Non-calculator1
FWork out 1,256,340 + 2,118,095.
3,374,435
Method. Use the column method, carrying where needed.
- Add the ones column. Write the ones digit of the total in this column, and carry the tens digit to the next column.0 + 5 + 0 = 5, write 5
- Add the tens column. Write the ones digit of the total in this column, and carry the tens digit to the next column.4 + 9 + 0 = 13, write 3, carry 1
- Add the hundreds column. Write the ones digit of the total in this column, and carry the tens digit to the next column.3 + 0 + 1 = 4, write 4
- Add the thousands column. Write the ones digit of the total in this column, and carry the tens digit to the next column.6 + 8 + 0 = 14, write 4, carry 1
- Add the ten thousands column. Write the ones digit of the total in this column, and carry the tens digit to the next column.5 + 1 + 1 = 7, write 7
- Add the hundred thousands column. Write the ones digit of the total in this column, and carry the tens digit to the next column.2 + 1 + 0 = 3, write 3
- Add the millions column. Write the ones digit of the total in this column, and carry the tens digit to the next column.1 + 2 + 0 = 3, write 3
- Write the answer. 3,374,435
Check. 2,118,095 + 1,256,340 = 3,374,435. ✓
2
FWork out 7,000,000 - 3,250,000.
3,750,000
Method. Subtract column by column, borrowing across the zeros.
- Subtract the ones column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.0 - 0 = 0
- Subtract the tens column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.0 - 0 = 0
- Subtract the hundreds column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.0 - 0 = 0
- Subtract the thousands column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.0 - 0 = 0
- Subtract the ten thousands column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.0 - 0 = 0, which is less than 5, so borrow 10: 10 - 5 = 5
- Subtract the hundred thousands column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.0 - 1 = -1, which is less than 2, so borrow 10: 9 - 2 = 7
- Subtract the millions column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.6 - 3 = 3
- Write the answer. 3,750,000
Check. 3,750,000 + 3,250,000 = 7,000,000. ✓
3
FWork out 45,678 + 9,876.
55,554
Method. Add column by column, carrying where needed.
- Add the ones column. Write the ones digit of the total in this column, and carry the tens digit to the next column.8 + 6 + 0 = 14, write 4, carry 1
- Add the tens column. Write the ones digit of the total in this column, and carry the tens digit to the next column.7 + 7 + 1 = 15, write 5, carry 1
- Add the hundreds column. Write the ones digit of the total in this column, and carry the tens digit to the next column.6 + 8 + 1 = 15, write 5, carry 1
- Add the thousands column. Write the ones digit of the total in this column, and carry the tens digit to the next column.5 + 9 + 1 = 15, write 5, carry 1
- Add the ten thousands column. Write the ones digit of the total in this column, and carry the tens digit to the next column.4 + 0 + 1 = 5, write 5
- Write the answer. 55,554
Check. Estimate: 46,000 + 10,000 = 56,000, close to 55,554. ✓
4
FWork out 100,000 - 37,426.
62,574
Method. Subtract column by column, borrowing.
- Subtract the ones column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.0 - 0 = 0, which is less than 6, so borrow 10: 10 - 6 = 4
- Subtract the tens column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.0 - 1 = -1, which is less than 2, so borrow 10: 9 - 2 = 7
- Subtract the hundreds column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.0 - 1 = -1, which is less than 4, so borrow 10: 9 - 4 = 5
- Subtract the thousands column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.0 - 1 = -1, which is less than 7, so borrow 10: 9 - 7 = 2
- Subtract the ten thousands column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.0 - 1 = -1, which is less than 3, so borrow 10: 9 - 3 = 6
- Subtract the hundred thousands column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.0 - 0 = 0
- Write the answer. 62,574
Check. 62,574 + 37,426 = 100,000. ✓
5
FWork out mentally: 3,004,000 + 996,000.
4,000,000
Method. Add the thousands in two parts.
- Round 996,000 up. 996,000 is 4,000 less than 1,000,000.3,004,000 + 1,000,000 = 4,004,000
- Take away 4,000. Subtract 4,000 from 4,004,000.4,004,000 - 4,000 = 4,000,000
- Write the answer. 4,000,000
Check. 4,000,000 - 996,000 = 3,004,000. ✓
6
FWork out mentally: 8,500,000 - 1,000,000.
7,500,000
Method. Change only the millions digit.
- Subtract 1 million. The millions digit goes from 8 to 7.8,500,000 - 1,000,000
- Write the answer. The rest of the number stays the same.7,500,000
- Write the answer. 7,500,000
Check. 7,500,000 + 1,000,000 = 8,500,000. ✓
7
FWork out mentally: 25,000 + 4,900.
29,900
Method. Add 5,000 and take away 100.
- Add 5,000. 25,000 + 5,000 = 30,000.30,000
- Take away 100. 4,900 is 100 less than 5,000.30,000 - 100 = 29,900
- Write the answer. 29,900
Check. 29,900 - 4,900 = 25,000. ✓
8
PSA museum had 2,345,678 visitors in one year and 1,987,654 the next. How many visitors were there in total?
4,333,332
Method. Add the two totals with a column method.
- Add the ones column. Write the ones digit of the total in this column, and carry the tens digit to the next column.8 + 4 + 0 = 12, write 2, carry 1
- Add the tens column. Write the ones digit of the total in this column, and carry the tens digit to the next column.7 + 5 + 1 = 13, write 3, carry 1
- Add the hundreds column. Write the ones digit of the total in this column, and carry the tens digit to the next column.6 + 6 + 1 = 13, write 3, carry 1
- Add the thousands column. Write the ones digit of the total in this column, and carry the tens digit to the next column.5 + 7 + 1 = 13, write 3, carry 1
- Add the ten thousands column. Write the ones digit of the total in this column, and carry the tens digit to the next column.4 + 8 + 1 = 13, write 3, carry 1
- Add the hundred thousands column. Write the ones digit of the total in this column, and carry the tens digit to the next column.3 + 9 + 1 = 13, write 3, carry 1
- Add the millions column. Write the ones digit of the total in this column, and carry the tens digit to the next column.2 + 1 + 1 = 4, write 4
- Write the answer. 4,333,332
Check. About 2,300,000 + 2,000,000 = 4,300,000, close to 4,333,332. ✓
9
PSA town has 3,450,000 people. 1,268,500 of them live in the north. How many people live elsewhere in the town?
2,181,500
Method. Subtract the number in the north from the total.
- Subtract the ones column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.0 - 0 = 0
- Subtract the tens column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.0 - 0 = 0
- Subtract the hundreds column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.0 - 0 = 0, which is less than 5, so borrow 10: 10 - 5 = 5
- Subtract the thousands column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.0 - 1 = -1, which is less than 8, so borrow 10: 9 - 8 = 1
- Subtract the ten thousands column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.5 - 1 = 4, which is less than 6, so borrow 10: 14 - 6 = 8
- Subtract the hundred thousands column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.3 - 2 = 1
- Subtract the millions column. Take the bottom digit from the top digit. If the top is too small, borrow 10 from the column on the left.3 - 1 = 2
- Write the answer. 2,181,500
Check. 2,181,500 + 1,268,500 = 3,450,000. ✓
10
PSSam has £10,000 in a savings account. He spends £3,678 on a bike. How much is left?
£6,322
Method. Subtract the cost from the savings.
- Write the calculation. Savings take away the cost.10,000 - 3,678
- Work it out. Subtract column by column with borrowing.10,000 - 3,678 = 6,322
- Write the answer. £6,322
Check. 6,322 + 3,678 = 10,000. ✓
11
RExplain how you would check 5,461,000 - 1,234,000 = 4,227,000 without working it out again.
Accept any answer that says add the answer back to the number that was taken away, and get the first number. For example: Add the answer to 1,234,000 and check you get 5,461,000.
Method. A subtraction is correct when the answer plus the smaller number gives the bigger number.
- Add the answer and the smaller number. Add 4,227,000 and 1,234,000.4,227,000 + 1,234,000 = 5,461,000
- Compare with the starting number. It is 5,461,000, which matches the first number.5,461,000 = 5,461,000
- Write the answer. Add the answer to 1,234,000 and check you get 5,461,000.
Check. The check works, so the subtraction is correct. ✓
12
RSam says 8,000,000 - 7,999,999 = 2. Is he right? Write the correct answer.
1
Method. Count on from 7,999,999 to 8,000,000.
- Count the steps. From 7,999,999 to 8,000,000 is one step.7,999,999 + 1 = 8,000,000
- Write the answer. The difference is 1, so Sam is wrong.8,000,000 - 7,999,999 = 1
- Write the answer. 1
Check. Sam is wrong: the answer is 1. ✓
Watch out. The gap between 7,999,999 and 8,000,000 is one, not two.