GCSE Maths: Adding and subtracting (Foundation)
Part of the GCSE Maths number basics chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Adding and subtracting
Addition and subtraction are inverse operations: one undoes the other. Because 47 + 38 = 85, we know that 85 − 38 = 47 and 85 − 47 = 38. The words sum, total, plus and increase mean add. The words difference, minus, less, decrease and “how many more” mean subtract.
Column addition. Write the numbers underneath each other so that digits of the same value are in the same column (units under units, tens under tens). Add the columns starting on the right. If a column adds up to 10 or more, write the units digit and carry the tens digit to the next column on the left.
Column subtraction. Put the bigger number on top and subtract each column, starting on the right. If the top digit is smaller than the bottom digit, exchange (also called borrowing): take 1 from the next column on the left and add it to this column as 10. If that column is 0, go further left.
Decimals. Line up the decimal points, not the right-hand ends of the numbers. Fill any gaps with zeros so that every number has the same number of decimal places: 5.2 + 0.37 is written 5.20 + 0.37. Then add or subtract as usual and put the decimal point in the answer straight under the others.
Mental methods. You can often avoid columns. To work out 47 + 38 you can partition 38 into 30 + 8: 47 + 30 = 77, then 77 + 8 = 85. To work out 63 + 49 you can compensate: add 50 then take 1 off: 63 + 50 = 113, 113 − 1 = 112. For a subtraction such as 100 − 63, you can count on from 63 up to 100: 63 + 7 = 70, then 70 + 30 = 100, so the difference is 7 + 30 = 37.
Always check. Use the inverse operation (add to check a subtraction, subtract to check an addition) or make a quick estimate by rounding. If the answer is not sensible, look for the mistake.
Key ideas
- Line up place values: units under units, decimal point under decimal point.
- Add from the right and carry. Subtract from the right and exchange when the top digit is too small.
- Fill gaps in decimals with zeros: 10 − 3.76 is 10.00 − 3.76.
- Check with the inverse operation: the answer to a subtraction plus the smaller number gives the bigger number.
Key words. The sum or total is the answer to an addition. Carry means move the tens digit of a column total to the next column on the left.
Method. Add using columns. Line up the place values, add from the right, and carry when a column reaches 10 or more.
- Line up the numbers in columns. Digits can only be added when they are worth the same, so units go under units, tens under tens, and so on.2684 + 1759 ------
- Add the columns, starting on the right. We start with the smallest column. If a column adds up to 10 or more, we write the units digit and carry the tens digit to the next column.Units: 4 + 9 = 13. Write 3 and carry 1 into the tens column.
Tens: 8 + 5 + 1 (carried) = 14. Write 4 and carry 1 into the hundreds column.
Hundreds: 6 + 7 + 1 (carried) = 14. Write 4 and carry 1 into the thousands column.
Thousands: 2 + 1 + 1 (carried) = 4. Write 4. - Write the answer. Read the digits of the bottom row. The small numbers on the bottom row show the carries.2684 + 1759 ------ 4443 111
2684 + 1759 = 4443
Check. Check with an inverse (opposite) operation: 4443 − 1759 = 2684, which is the first number. ✓
Watch out. Do not forget to add the carried digit into the next column.
Key words. The difference is the answer to a subtraction. Exchange (or borrowing) means taking 1 from a column and putting 10 into the column on its right.
Method. Subtract using columns. Line up the place values, subtract from the right, and exchange when the top digit is too small.
- Line up the numbers in columns, with the bigger number on top. Digits can only be subtracted when they are worth the same: units under units, tens under tens.5003 − 1478 ------
- Subtract each column, starting on the right. If the top digit is smaller than the digit below it, we exchange (borrow): take 1 from the column on the left and add it to this column as 10.Units: 3 is smaller than 8, so we must exchange. The tens and hundreds columns are 0, so there is nothing to take there. Go left to the thousands column. Take 1 thousand (5 becomes 4) and exchange it for 10 hundreds. Take 1 of those hundreds (leaving 9) and exchange it for 10 tens. Take 1 of those tens (leaving 9) and exchange it for 10 units. 3 becomes 13. Now 13 − 8 = 5.
Tens: 9 − 7 = 2.
Hundreds: 9 − 4 = 5.
Thousands: 4 − 1 = 3. - Write the answer. Read the bottom row.5003 − 1478 ------ 3525
5003 − 1478 = 3525
Check. Check with the inverse (opposite) operation, addition: 3525 + 1478 = 5003. ✓
Watch out. Always subtract the smaller number from the bigger in each column only after exchanging. Never take the top digit from the bottom one.
Key words. The difference is the answer to a subtraction. Exchange (or borrowing) means taking 1 from a column and putting 10 into the column on its right.
Method. Subtract using columns. Line up the place values, subtract from the right, and exchange when the top digit is too small.
- Line up the numbers in columns, with the bigger number on top. Digits can only be subtracted when they are worth the same: units under units, tens under tens, decimal points under decimal points. Fill gaps with zeros.10.00 − 3.76 -------
- Subtract each column, starting on the right. If the top digit is smaller than the digit below it, we exchange (borrow): take 1 from the column on the left and add it to this column as 10.Hundredths: 0 is smaller than 6, so we must exchange. The tenths and units columns are 0, so there is nothing to take there. Go left to the tens column. Take 1 ten (1 becomes 0) and exchange it for 10 units. Take 1 of those units (leaving 9) and exchange it for 10 tenths. Take 1 of those tenths (leaving 9) and exchange it for 10 hundredths. 0 becomes 10. Now 10 − 6 = 4.
Tenths: 9 − 7 = 2.
Units: 9 − 3 = 6.
Tens: 0 − 0 = 0. - Write the answer. Put the decimal point in line with the others.10.00 − 3.76 ------- 06.24
10 − 3.76 = 6.24
Check. Check with the inverse (opposite) operation, addition: 6.24 + 3.76 = 10. ✓
Watch out. Always subtract the smaller number from the bigger in each column only after exchanging. Never take the top digit from the bottom one.
Watch out
Line up the decimal points, not the ends of the numbers. Writing 5.2 + 0.37 with the 2 above the 7 gives a wrong answer. Write 5.20 + 0.37 so that the tenths are under the tenths.Exercise 1B Adding and subtracting
Non-calculatorKey words. The sum or total is the answer to an addition. Carry means move the tens digit of a column total to the next column on the left.
Method. Add using columns. Line up the place values, add from the right, and carry when a column reaches 10 or more.
- Line up the numbers in columns. Digits can only be added when they are worth the same, so units go under units, tens under tens, and so on.345 + 278 -----
- Add the columns, starting on the right. We start with the smallest column. If a column adds up to 10 or more, we write the units digit and carry the tens digit to the next column.Units: 5 + 8 = 13. Write 3 and carry 1 into the tens column.
Tens: 4 + 7 + 1 (carried) = 12. Write 2 and carry 1 into the hundreds column.
Hundreds: 3 + 2 + 1 (carried) = 6. Write 6. - Write the answer. Read the digits of the bottom row. The small numbers on the bottom row show the carries.345 + 278 ----- 623 11
345 + 278 = 623
Check. Check with an inverse (opposite) operation: 623 − 278 = 345, which is the first number. ✓
Watch out. Do not forget to add the carried digit into the next column.
Key words. The sum or total is the answer to an addition. Carry means move the tens digit of a column total to the next column on the left.
Method. Add using columns. Line up the place values, add from the right, and carry when a column reaches 10 or more.
- Line up the numbers in columns. Digits can only be added when they are worth the same, so units go under units, tens under tens, and so on.4806 + 1975 ------
- Add the columns, starting on the right. We start with the smallest column. If a column adds up to 10 or more, we write the units digit and carry the tens digit to the next column.Units: 6 + 5 = 11. Write 1 and carry 1 into the tens column.
Tens: 0 + 7 + 1 (carried) = 8. Write 8.
Hundreds: 8 + 9 = 17. Write 7 and carry 1 into the thousands column.
Thousands: 4 + 1 + 1 (carried) = 6. Write 6. - Write the answer. Read the digits of the bottom row. The small numbers on the bottom row show the carries.4806 + 1975 ------ 6781 1 1
4806 + 1975 = 6781
Check. Check with an inverse (opposite) operation: 6781 − 1975 = 4806, which is the first number. ✓
Watch out. Do not forget to add the carried digit into the next column.
Key words. The sum or total is the answer to an addition. Carry means move the tens digit of a column total to the next column on the left.
Method. Add using columns. Line up the place values, add from the right, and carry when a column reaches 10 or more.
- Line up the numbers in columns. Digits can only be added when they are worth the same, so units go under units, tens under tens, and decimal points under decimal points. Fill any gaps with zeros.73.40 + 8.95 -------
- Add the columns, starting on the right. We start with the smallest column. If a column adds up to 10 or more, we write the units digit and carry the tens digit to the next column.Hundredths: 0 + 5 = 5. Write 5.
Tenths: 4 + 9 = 13. Write 3 and carry 1 into the units column.
Units: 3 + 8 + 1 (carried) = 12. Write 2 and carry 1 into the tens column.
Tens: 7 + 0 + 1 (carried) = 8. Write 8. - Put the decimal point in the answer. The point goes directly under the other decimal points.73.40 + 8.95 ------- 82.35 11
73.4 + 8.95 = 82.35
Check. Check with an inverse (opposite) operation: 82.35 − 8.95 = 73.4, which is the first number. ✓
Watch out. Line up the decimal points, not the right-hand ends of the numbers. 5.2 + 0.37 needs 5.20 + 0.37.
Key words. The sum or total is the answer to an addition. Carry means move the tens digit of a column total to the next column on the left.
Method. Add using columns. Line up the place values, add from the right, and carry when a column reaches 10 or more.
- Line up the numbers in columns. Digits can only be added when they are worth the same, so units go under units, tens under tens, and so on.256 389 + 47 -----
- Add the columns, starting on the right. We start with the smallest column. If a column adds up to 10 or more, we write the units digit and carry the tens digit to the next column.Units: 6 + 9 + 7 = 22. Write 2 and carry 2 into the tens column.
Tens: 5 + 8 + 4 + 2 (carried) = 19. Write 9 and carry 1 into the hundreds column.
Hundreds: 2 + 3 + 0 + 1 (carried) = 6. Write 6. - Write the answer. Read the digits of the bottom row. The small numbers on the bottom row show the carries.256 389 + 47 ----- 692 12
256 + 389 + 47 = 692
Check. Check with an inverse (opposite) operation: 692 − 47 = 645 = 645, which is 256 + 389. ✓
Watch out. Do not forget to add the carried digit into the next column.
Key words. The sum or total is the answer to an addition. Carry means move the tens digit of a column total to the next column on the left.
Method. Add using columns. Line up the place values, add from the right, and carry when a column reaches 10 or more.
- Line up the numbers in columns. Digits can only be added when they are worth the same, so units go under units, tens under tens, and decimal points under decimal points. Fill any gaps with zeros.0.68 + 5.70 ------
- Add the columns, starting on the right. We start with the smallest column. If a column adds up to 10 or more, we write the units digit and carry the tens digit to the next column.Hundredths: 8 + 0 = 8. Write 8.
Tenths: 6 + 7 = 13. Write 3 and carry 1 into the units column.
Units: 0 + 5 + 1 (carried) = 6. Write 6. - Put the decimal point in the answer. The point goes directly under the other decimal points.0.68 + 5.70 ------ 6.38 1
0.68 + 5.7 = 6.38
Check. Check with an inverse (opposite) operation: 6.38 − 5.7 = 0.68, which is the first number. ✓
Watch out. Line up the decimal points, not the right-hand ends of the numbers. 5.2 + 0.37 needs 5.20 + 0.37.
Key words. The sum or total is the answer to an addition. Carry means move the tens digit of a column total to the next column on the left.
Method. Add using columns. Line up the place values, add from the right, and carry when a column reaches 10 or more.
- Line up the numbers in columns. Digits can only be added when they are worth the same, so units go under units, tens under tens, and so on.12480 + 9735 -------
- Add the columns, starting on the right. We start with the smallest column. If a column adds up to 10 or more, we write the units digit and carry the tens digit to the next column.Units: 0 + 5 = 5. Write 5.
Tens: 8 + 3 = 11. Write 1 and carry 1 into the hundreds column.
Hundreds: 4 + 7 + 1 (carried) = 12. Write 2 and carry 1 into the thousands column.
Thousands: 2 + 9 + 1 (carried) = 12. Write 2 and carry 1 into the ten thousands column.
Ten thousands: 1 + 0 + 1 (carried) = 2. Write 2. - Write the answer. Read the digits of the bottom row. The small numbers on the bottom row show the carries.12480 + 9735 ------- 22215 111
12 480 + 9735 = 22 215
Check. Check with an inverse (opposite) operation: 22 215 − 9735 = 12 480, which is the first number. ✓
Watch out. Do not forget to add the carried digit into the next column.
Key words. The difference is the answer to a subtraction. Exchange (or borrowing) means taking 1 from a column and putting 10 into the column on its right.
Method. Subtract using columns. Line up the place values, subtract from the right, and exchange when the top digit is too small.
- Line up the numbers in columns, with the bigger number on top. Digits can only be subtracted when they are worth the same: units under units, tens under tens.802 − 457 -----
- Subtract each column, starting on the right. If the top digit is smaller than the digit below it, we exchange (borrow): take 1 from the column on the left and add it to this column as 10.Units: 2 is smaller than 7, so we must exchange. The tens column is 0, so there is nothing to take there. Go left to the hundreds column. Take 1 hundred (8 becomes 7) and exchange it for 10 tens. Take 1 of those tens (leaving 9) and exchange it for 10 units. 2 becomes 12. Now 12 − 7 = 5.
Tens: 9 − 5 = 4.
Hundreds: 7 − 4 = 3. - Write the answer. Read the bottom row.802 − 457 ----- 345
802 − 457 = 345
Check. Check with the inverse (opposite) operation, addition: 345 + 457 = 802. ✓
Watch out. Always subtract the smaller number from the bigger in each column only after exchanging. Never take the top digit from the bottom one.
Key words. The difference is the answer to a subtraction. Exchange (or borrowing) means taking 1 from a column and putting 10 into the column on its right.
Method. Subtract using columns. Line up the place values, subtract from the right, and exchange when the top digit is too small.
- Line up the numbers in columns, with the bigger number on top. Digits can only be subtracted when they are worth the same: units under units, tens under tens.6000 − 2348 ------
- Subtract each column, starting on the right. If the top digit is smaller than the digit below it, we exchange (borrow): take 1 from the column on the left and add it to this column as 10.Units: 0 is smaller than 8, so we must exchange. The tens and hundreds columns are 0, so there is nothing to take there. Go left to the thousands column. Take 1 thousand (6 becomes 5) and exchange it for 10 hundreds. Take 1 of those hundreds (leaving 9) and exchange it for 10 tens. Take 1 of those tens (leaving 9) and exchange it for 10 units. 0 becomes 10. Now 10 − 8 = 2.
Tens: 9 − 4 = 5.
Hundreds: 9 − 3 = 6.
Thousands: 5 − 2 = 3. - Write the answer. Read the bottom row.6000 − 2348 ------ 3652
6000 − 2348 = 3652
Check. Check with the inverse (opposite) operation, addition: 3652 + 2348 = 6000. ✓
Watch out. Always subtract the smaller number from the bigger in each column only after exchanging. Never take the top digit from the bottom one.
Key words. The difference is the answer to a subtraction. Exchange (or borrowing) means taking 1 from a column and putting 10 into the column on its right.
Method. Subtract using columns. Line up the place values, subtract from the right, and exchange when the top digit is too small.
- Line up the numbers in columns, with the bigger number on top. Digits can only be subtracted when they are worth the same: units under units, tens under tens, decimal points under decimal points. Fill gaps with zeros.15.20 − 7.85 -------
- Subtract each column, starting on the right. If the top digit is smaller than the digit below it, we exchange (borrow): take 1 from the column on the left and add it to this column as 10.Hundredths: 0 is smaller than 5, so we cannot take 5 from 0. Exchange: take 1 tenth from the tenths column (2 becomes 1) and add it to the hundredths column as 10 hundredths. 0 becomes 10. Now 10 − 5 = 5.
Tenths: 1 is smaller than 8, so we cannot take 8 from 1. Exchange: take 1 one from the units column (5 becomes 4) and add it to the tenths column as 10 tenths. 1 becomes 11. Now 11 − 8 = 3.
Units: 4 is smaller than 7, so we cannot take 7 from 4. Exchange: take 1 ten from the tens column (1 becomes 0) and add it to the units column as 10 units. 4 becomes 14. Now 14 − 7 = 7.
Tens: 0 − 0 = 0. - Write the answer. Put the decimal point in line with the others.15.20 − 7.85 ------- 07.35
15.2 − 7.85 = 7.35
Check. Check with the inverse (opposite) operation, addition: 7.35 + 7.85 = 15.2. ✓
Watch out. Always subtract the smaller number from the bigger in each column only after exchanging. Never take the top digit from the bottom one.
Key words. The difference is the answer to a subtraction. Exchange (or borrowing) means taking 1 from a column and putting 10 into the column on its right.
Method. Subtract using columns. Line up the place values, subtract from the right, and exchange when the top digit is too small.
- Line up the numbers in columns, with the bigger number on top. Digits can only be subtracted when they are worth the same: units under units, tens under tens, decimal points under decimal points. Fill gaps with zeros.9.00 − 3.64 ------
- Subtract each column, starting on the right. If the top digit is smaller than the digit below it, we exchange (borrow): take 1 from the column on the left and add it to this column as 10.Hundredths: 0 is smaller than 4, so we must exchange. The tenths column is 0, so there is nothing to take there. Go left to the units column. Take 1 one (9 becomes 8) and exchange it for 10 tenths. Take 1 of those tenths (leaving 9) and exchange it for 10 hundredths. 0 becomes 10. Now 10 − 4 = 6.
Tenths: 9 − 6 = 3.
Units: 8 − 3 = 5. - Write the answer. Put the decimal point in line with the others.9.00 − 3.64 ------ 5.36
9 − 3.64 = 5.36
Check. Check with the inverse (opposite) operation, addition: 5.36 + 3.64 = 9. ✓
Watch out. Always subtract the smaller number from the bigger in each column only after exchanging. Never take the top digit from the bottom one.
Key words. The difference is the answer to a subtraction. Exchange (or borrowing) means taking 1 from a column and putting 10 into the column on its right.
Method. Subtract using columns. Line up the place values, subtract from the right, and exchange when the top digit is too small.
- Line up the numbers in columns, with the bigger number on top. Digits can only be subtracted when they are worth the same: units under units, tens under tens.30105 − 8967 -------
- Subtract each column, starting on the right. If the top digit is smaller than the digit below it, we exchange (borrow): take 1 from the column on the left and add it to this column as 10.Units: 5 is smaller than 7, so we must exchange. The tens column is 0, so there is nothing to take there. Go left to the hundreds column. Take 1 hundred (1 becomes 0) and exchange it for 10 tens. Take 1 of those tens (leaving 9) and exchange it for 10 units. 5 becomes 15. Now 15 − 7 = 8.
Tens: 9 − 6 = 3.
Hundreds: 0 is smaller than 9, so we must exchange. The thousands column is 0, so there is nothing to take there. Go left to the ten thousands column. Take 1 ten thousand (3 becomes 2) and exchange it for 10 thousands. Take 1 of those thousands (leaving 9) and exchange it for 10 hundreds. 0 becomes 10. Now 10 − 9 = 1.
Thousands: 9 − 8 = 1.
Ten thousands: 2 − 0 = 2. - Write the answer. Read the bottom row.30105 − 8967 ------- 21138
30 105 − 8967 = 21 138
Check. Check with the inverse (opposite) operation, addition: 21 138 + 8967 = 30 105. ✓
Watch out. Always subtract the smaller number from the bigger in each column only after exchanging. Never take the top digit from the bottom one.
Key words. The difference is the answer to a subtraction. Exchange (or borrowing) means taking 1 from a column and putting 10 into the column on its right.
Method. Subtract using columns. Line up the place values, subtract from the right, and exchange when the top digit is too small.
- Line up the numbers in columns, with the bigger number on top. Digits can only be subtracted when they are worth the same: units under units, tens under tens, decimal points under decimal points. Fill gaps with zeros.4.200 − 0.375 -------
- Subtract each column, starting on the right. If the top digit is smaller than the digit below it, we exchange (borrow): take 1 from the column on the left and add it to this column as 10.Thousandths: 0 is smaller than 5, so we must exchange. The hundredths column is 0, so there is nothing to take there. Go left to the tenths column. Take 1 tenth (2 becomes 1) and exchange it for 10 hundredths. Take 1 of those hundredths (leaving 9) and exchange it for 10 thousandths. 0 becomes 10. Now 10 − 5 = 5.
Hundredths: 9 − 7 = 2.
Tenths: 1 is smaller than 3, so we cannot take 3 from 1. Exchange: take 1 one from the units column (4 becomes 3) and add it to the tenths column as 10 tenths. 1 becomes 11. Now 11 − 3 = 8.
Units: 3 − 0 = 3. - Write the answer. Put the decimal point in line with the others.4.200 − 0.375 ------- 3.825
4.2 − 0.375 = 3.825
Check. Check with the inverse (opposite) operation, addition: 3.825 + 0.375 = 4.2. ✓
Watch out. Always subtract the smaller number from the bigger in each column only after exchanging. Never take the top digit from the bottom one.
Method. Partition the second number into hundreds, tens and ones and add the parts one at a time.
- Split the second number into hundreds, tens and ones. Smaller numbers are easier to work with in your head. We can add or subtract the parts one at a time.38 = 30 + 8
- Add each part in turn, biggest part first. Keep the running total each time and do not lose track of it.47 + 30 = 77
77 + 8 = 85 - Write the answer. 47 + 38 = 85
Check. Reverse: 85 − 38 = 47. ✓
Watch out. Keep a running total. Do not mix up the tens and the ones.
Method. Round the second number to a multiple of 10, do the easy calculation, then correct for the amount you changed it by.
- Round 49 to a nearby multiple of 10. 50 is 1 more than 49, and 50 is easy to add.49 + 1 = 50
- Add the rounded number. This is an easy calculation.63 + 50 = 113
- Adjust the answer. We added 1 too much, so we take it back off.113 − 1 = 112
Check. Reverse: 112 − 49 = 63. ✓
Watch out. Say which way you rounded. If you rounded up, you used too much, so the adjustment goes the opposite way.
Method. Partition the second number into hundreds, tens and ones and subtract the parts one at a time.
- Split the second number into hundreds, tens and ones. Smaller numbers are easier to work with in your head. We can add or subtract the parts one at a time.35 = 30 + 5
- Subtract each part in turn, biggest part first. Keep the running total each time and do not lose track of it.152 − 30 = 122
122 − 5 = 117 - Write the answer. 152 − 35 = 117
Check. Reverse: 117 + 35 = 152. ✓
Watch out. Keep a running total. Do not mix up the tens and the ones.
Method. Round the second number to a multiple of 10, do the easy calculation, then correct for the amount you changed it by.
- Round 198 to a nearby multiple of 10. 200 is 2 more than 198, and 200 is easy to subtract.198 + 2 = 200
- Subtract the rounded number. This is an easy calculation.205 − 200 = 5
- Adjust the answer. We subtracted 2 too much, so we put it back.5 + 2 = 7
Check. Reverse: 7 + 198 = 205. ✓
Watch out. Say which way you rounded. If you rounded up, you used too much, so the adjustment goes the opposite way.
Key words. The sum or total is the answer to an addition. Carry means move the tens digit of a column total to the next column on the left.
Method. Add using columns. Line up the place values, add from the right, and carry when a column reaches 10 or more.
- Line up the numbers in columns. Digits can only be added when they are worth the same, so units go under units, tens under tens, and so on.468 + 512 -----
- Add the columns, starting on the right. We start with the smallest column. If a column adds up to 10 or more, we write the units digit and carry the tens digit to the next column.Units: 8 + 2 = 10. Write 0 and carry 1 into the tens column.
Tens: 6 + 1 + 1 (carried) = 8. Write 8.
Hundreds: 4 + 5 = 9. Write 9. - Write the answer. Read the digits of the bottom row. The small numbers on the bottom row show the carries.468 + 512 ----- 980 1
468 + 512 = 980
Check. Check with an inverse (opposite) operation: 980 − 512 = 468, which is the first number. ✓
Watch out. Do not forget to add the carried digit into the next column.
Key words. The difference is the answer to a subtraction. Exchange (or borrowing) means taking 1 from a column and putting 10 into the column on its right.
Method. Subtract using columns. Line up the place values, subtract from the right, and exchange when the top digit is too small.
- Line up the numbers in columns, with the bigger number on top. Digits can only be subtracted when they are worth the same: units under units, tens under tens.1250 − 876 ------
- Subtract each column, starting on the right. If the top digit is smaller than the digit below it, we exchange (borrow): take 1 from the column on the left and add it to this column as 10.Units: 0 is smaller than 6, so we cannot take 6 from 0. Exchange: take 1 ten from the tens column (5 becomes 4) and add it to the units column as 10 units. 0 becomes 10. Now 10 − 6 = 4.
Tens: 4 is smaller than 7, so we cannot take 7 from 4. Exchange: take 1 hundred from the hundreds column (2 becomes 1) and add it to the tens column as 10 tens. 4 becomes 14. Now 14 − 7 = 7.
Hundreds: 1 is smaller than 8, so we cannot take 8 from 1. Exchange: take 1 thousand from the thousands column (1 becomes 0) and add it to the hundreds column as 10 hundreds. 1 becomes 11. Now 11 − 8 = 3.
Thousands: 0 − 0 = 0. - Write the answer. Read the bottom row.1250 − 876 ------ 0374
1250 − 876 = 374
Check. Check with the inverse (opposite) operation, addition: 374 + 876 = 1250. ✓
Watch out. Always subtract the smaller number from the bigger in each column only after exchanging. Never take the top digit from the bottom one.
Method. Subtraction and addition are inverses. If 3004 − 1856 = 1248 then 1248 + 1856 should give 3004.
- Add her answer to the number that was subtracted. If she is right, we will get back to the number she started with.1248 + 1856 = 3104
- Compare with the starting number. 3104 is not 3004, so her answer is wrong.3104 ≠ 3004
- Find the correct answer. Use column subtraction with exchanging across the zeros.3004 − 1856 = 1148
Check. 1148 + 1856 = 3004. ✓
Watch out. A check using the inverse operation will always show up an error. Do it every time.
Method. Ali has not lined up the decimal points. Line them up and add again.
- Describe the mistake. He added 52 and 37 as if the digits lined up at the right, so he added tenths to hundredths. 5.2 + 0.37 cannot be less than 5.His answer 0.89 is smaller than 5.2, which cannot be right for an addition.
- Write the numbers with the decimal points lined up and the gaps filled with a zero. Then tenths are under tenths and hundredths under hundredths.5.20
+ 0.37 - Add. Hundredths: 0 + 7 = 7. Tenths: 2 + 3 = 5. Units: 5 + 0 = 5.5.20 + 0.37 = 5.57
Check. 5.57 − 0.37 = 5.20. ✓
Key words. The sum or total is the answer to an addition. Carry means move the tens digit of a column total to the next column on the left.
Method. Add using columns. Line up the place values, add from the right, and carry when a column reaches 10 or more.
- Line up the numbers in columns. Digits can only be added when they are worth the same, so units go under units, tens under tens, and decimal points under decimal points. Fill any gaps with zeros.3.45 12.99 + 0.85 -------
- Add the columns, starting on the right. We start with the smallest column. If a column adds up to 10 or more, we write the units digit and carry the tens digit to the next column.Hundredths: 5 + 9 + 5 = 19. Write 9 and carry 1 into the tenths column.
Tenths: 4 + 9 + 8 + 1 (carried) = 22. Write 2 and carry 2 into the units column.
Units: 3 + 2 + 0 + 2 (carried) = 7. Write 7.
Tens: 0 + 1 + 0 = 1. Write 1. - Put the decimal point in the answer. The point goes directly under the other decimal points.3.45 12.99 + 0.85 ------- 17.29 2 1
3.45 + 12.99 + 0.85 = 17.29
Check. Check with an inverse (opposite) operation: 17.29 − 0.85 = 16.44 = 16.44, which is 3.45 + 12.99. ✓
Watch out. Line up the decimal points, not the right-hand ends of the numbers. 5.2 + 0.37 needs 5.20 + 0.37.
Key words. The difference is the answer to a subtraction. Exchange (or borrowing) means taking 1 from a column and putting 10 into the column on its right.
Method. Subtract using columns. Line up the place values, subtract from the right, and exchange when the top digit is too small.
- Line up the numbers in columns, with the bigger number on top. Digits can only be subtracted when they are worth the same: units under units, tens under tens, decimal points under decimal points. Fill gaps with zeros.20.00 − 17.29 -------
- Subtract each column, starting on the right. If the top digit is smaller than the digit below it, we exchange (borrow): take 1 from the column on the left and add it to this column as 10.Hundredths: 0 is smaller than 9, so we must exchange. The tenths and units columns are 0, so there is nothing to take there. Go left to the tens column. Take 1 ten (2 becomes 1) and exchange it for 10 units. Take 1 of those units (leaving 9) and exchange it for 10 tenths. Take 1 of those tenths (leaving 9) and exchange it for 10 hundredths. 0 becomes 10. Now 10 − 9 = 1.
Tenths: 9 − 2 = 7.
Units: 9 − 7 = 2.
Tens: 1 − 1 = 0. - Write the answer. Put the decimal point in line with the others.20.00 − 17.29 ------- 02.71
20 − 17.29 = 2.71
Check. Check with the inverse (opposite) operation, addition: 2.71 + 17.29 = 20. ✓
Watch out. Always subtract the smaller number from the bigger in each column only after exchanging. Never take the top digit from the bottom one.
Method. Work through the columns from the right. The hidden digits are found by working backwards with carries.
- Look at the units column. 7 plus the hidden digit must end in 3. The only digit that works is 6, because 7 + 6 = 13. Write 3 and carry 1.7 + 6 = 13
- Look at the tens column. The tens column holds the carry of 1, the hidden digit, and 8. Together they must make a number ending in 0 (the tens digit of 703), so they add to 10 and carry 1.hidden + 8 + 1 = 10, so hidden = 1
- Check the hundreds column. The carry 1 plus 4 plus 2 should be 7.1 + 4 + 2 = 7 ✓
- Write the sum. The full calculation is:417 + 286 = 703
Check. 417 + 286 = 703. ✓
Method. Think about which columns are worth the most. Put the biggest digits in the biggest columns.
- Think about the columns. A hundreds digit is worth 100 times a units digit, so the biggest digits must go in the hundreds columns, the next biggest in the tens columns, and the smallest in the units columns.Hundreds: 6 and 5. Tens: 4 and 3. Units: 2 and 1.
- Write one possible pair of numbers. The sum is the same however the pairs are shared out, because each column adds to the same total.642 + 531
- Add. Units: 2 + 1 = 3. Tens: 4 + 3 = 7. Hundreds: 6 + 5 = 11.642 + 531 = 1173
Check. Another pair, 641 + 532, also gives 1173. Any swap within a column keeps the same total. ✓