A subtraction is a gap
Grandma is 83 and her grandson is 29. How much older is she? The answer is 83 − 29, the gap between the two ages. On a number line it is the distance from the mark at 29 to the mark at 83.
83 − 29 is the distance between the two marks.
Move both, keep the gap
Next year Grandma will be 84 and her grandson will be 30. Both are one year older, so the gap between their ages has not changed: 84 − 30 is the same as 83 − 29.
84 − 30 is easy, because 30 is a round number: 84 − 30 = 54. So 83 − 29 = 54 as well. Moving both numbers up by 1 turned the awkward 29 into 30.
This works for any subtraction. Add the same amount to both numbers, or take the same amount from both, and the difference stays the same. Choose the amount that makes the number being taken away a round number.
Each end hops up by 1: 29 to 30, and 83 to 84. Both hops are the same length, so the gap between the ends does not change.
The marks now sit at 30 and 84, the same distance apart as before: 84 − 30 = 54.
Slide up or slide down
For 83 − 47, the nearest round number to 47 is 50, which is 3 more. Add 3 to both numbers: 83 − 47 = 86 − 50 = 36. Or take 7 from both to reach 40: 83 − 47 = 76 − 40 = 36. Both slides give 36, because neither one changed the gap.
slide both by +0: 83 − 47 = 36, the same gap, because adding d to both ends of a subtraction cancels, (83 + d) − (47 + d) = 36
Slide both ends until the number being subtracted is a multiple of 10
The gold bar runs from 47 to 83, and it is 36 long. Sliding moves both ends together, so the bar keeps its length while the numbers at its ends change.
Both numbers, the same way
Two slips break the rule. Moving only one number changes the gap: 83 − 30 = 53, not 54, because only the 29 moved. And adding to one number while taking the same amount from the other changes the gap by twice that amount: 84 − 28 = 56.
That second move belongs to adding, where it keeps the total the same: 58 + 29 = 57 + 30 = 87. For a difference, both numbers move the same way by the same amount.
Worked example: Two Savings That Grow by the Same Amount
Question Sam has saved $503 and Lily has saved $298. Grandma gives each child $2 more. (a) How much does each child have now? (b) How much more does Sam have than Lily?
1.(a) 503 + 2 = 505 and 298 + 2 = 300. Sam has $505 and Lily has $300 now.
(a) 503 + 2 = 505 and 298 + 2 = 300. 2.Both children were given the same amount, so the difference between their savings has not changed.
Both bars grew by the same $2, so the difference between them has not changed. 3.Subtract the new numbers, because 300 is easy to take away: 505 − 300 = 205.
300 is easy to take away: 505 − 300 = 205. 4.(b) Sam has $205 more than Lily. This means 503 − 298 = 205 as well. Check: 298 + 205 = 503.
(b) Sam has $205 more than Lily.
Answer: (a) Sam has $505 and Lily has $300; (b) $205
Common mistakes
- Changing only 298 to 300 and working out 503 − 300 = 203. The difference stays the same only when both numbers move by the same amount.
- Adding 2 to one number and taking 2 from the other. That keeps the total the same, not the difference. For a difference, both numbers move the same way.
More adding and subtracting in your head problems, worked step by step →