Subtract by adding up
Stop taking numbers away. Walk up from the smaller number to the bigger one and count the steps. It makes 1000 − 637 a three-second job and kills borrowing for good.
THE IDEA
A subtraction is a distance
a − b is just the gap between b and a on the number line. A gap doesn't care which end you start from, so instead of asking 'what's left after I take 637 away?' you can ask 'how far is it from 637 up to 1000?' Same number, and walking up never needs a borrow.
1000 − 637 becomes: from 637, how do I get to 1000? A small hop to 640, a bigger one to 700, then a clean 300.
1000 − 637 (you paid 1000 for something at 637)
637 → 640 +3
640 → 700 +60
700 → 1000 +300
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add the hops: 3 + 60 + 300363Always hop to the next round number first, then the next hundred, then the finish. Those three sizes of hop cover almost every change-from-a-note problem you will meet.
Check yourself
Negar hands over 1000 for a book priced 458. Count up. What's her change?
- 552
- 540
- 458
- 542
Show the answer
542
Yes. 458 → 460 is 2, 460 → 500 is 40, 500 → 1000 is 500. That's 2 + 40 + 500 = 542.
SHORTCUT
Complements: what's missing to 100 or 1000
For 100 − 37, think of 100 as 99 + 1. Subtracting from 99 never borrows: 9 − 3 = 6 and 9 − 7 = 2, giving 62, then add the spare 1 to get 63. In practice people say it as a rhythm: digits pair to 9, the last pair to 10. For 1000 − 637: 9−6 = 3, 9−3 = 6, 10−7 = 3, so 363.
100 − 37 = 63. 1000 − 637 = 363. Same answer the hops gave you, in one breath.
Check yourself
What's missing? Match each gap to its size.
Show the answer
- 100 − 28 → 72
- 100 − 65 → 35
- 1000 − 245 → 755
- 1000 − 807 → 193
Two ways to face a subtraction
Take away
Good when the numbers are far apart and the second one is small: 863 − 12. Keep 863 whole and remove 12. Counting up here would mean 851 tiny steps of thinking.
Count up
Good when the numbers are close together, or when the bigger one is round: 1000 − 637, 2026 − 1987, 500 − 468. The gap is small, so you walk it in two or three hops.
Check yourself
- 1000 − 992: from 992 it's one hop of 8 to reach 1000. Two seconds.
- 1000 − 112: from 112 you'd have to walk 888, in several hops. Taking 112 away from 1000 in pieces is quicker.
What decides whether counting up is the faster way?
- How close the two numbers are to each other
- Whether the bigger number ends in zeros
- Whether the answer will be an even number
Show the answer
How close the two numbers are to each other
Right. Counting up wins when the gap is short, because you are walking the answer itself. A long gap means a long walk.
Bigger money, same three hops
- The problem: 2,000,000 − 1,365,000
Rent paid out of savings. Drop the three zeros in your head and work with 2,000 − 1,365; you put the zeros back at the end.
- Hop to the round hundred: 1,365 → 1,400
That's 35.
- Hop to the finish: 1,400 → 2,000
That's 600.
- Add the hops: 35 + 600 = 635
Zeros back on: 635,000 left. Two hops, no borrowing, no paper.
Check yourself
Reza pays with a 500 note for something that costs 317. He should get more than 200 back.
Show the answer
False
False. From 317 it's 3 up to 320, 80 up to 400, and 100 up to 500, so the change is 183. A quick sanity check gets you there too: 317 is more than 300, so the change has to be less than 200.
Make it a habit
- Say the question differently: not 'take 637 away' but 'from 637, how far to 1000?'
- Hop to the next ten, then the next hundred, then the target. Three hops covers most of life.
- Learn the pairs that make 100 by heart: 28 and 72, 35 and 65, 47 and 53. They come up constantly.
- Counting up can never borrow, which is why it's the method that survives a noisy shop.
Check yourself
The hard one. A 3000 budget, 1847 already spent. How much is left?
- 1,253
- 1,150
- 1,153
- 1,163
Show the answer
1,153
Exactly. 1847 → 1850 is 3, → 1900 is 50, → 2000 is 100, → 3000 is 1000. That's 3 + 50 + 100 + 1000 = 1,153.
Lesson recap
- A subtraction is the gap between two numbers, so you may walk it upwards instead of downwards.
- Hop to the next ten, then the next hundred, then the target, and add the hops.
- Complements: 100 − 37 = 63 and 1000 − 637 = 363, because the digits pair to 9 and the last pair to 10.
- That digit rhythm fails when the number ends in 0, so fall back on counting up.