A subtraction is a gap
63 − 58 asks how far apart 58 and 63 are. The two numbers are close together, so taking 58 away from 63 is the long way round. Start at the smaller number, 58, and count up to 63 instead.
Hop to the ten, then on
Count up in hops that are easy to add. From 58, 2 more reaches 60, a round number. From 60, 3 more reaches 63. The hops are 2 and 3, so the gap is 2 + 3 = 5, and 63 − 58 = 5.
Bigger gaps work the same way. For 82 − 67, it is 3 from 67 to 70, then 12 from 70 to 82, so 82 − 67 = 3 + 12 = 15. Adding checks it: 67 + 15 = 82.
Counting up also gets round the zeros that make column subtraction hard. For 1000 − 674, it is 26 from 674 to 700, then 300 from 700 to 1000, so 1000 − 674 = 326.
From 58, a hop of 2 reaches 60 and a hop of 3 reaches 63, so the gap is 5.
Counting out change
Shopkeepers count change this way. Something costs 45 cents and is paid for with a dollar, which is 100 cents. The shopkeeper counts up from the price: a 5-cent coin makes 50, a 25-cent coin makes 75, and another 25-cent coin makes 100. The coins handed over are the change, 5 + 25 + 25 = 55 cents, and nothing was taken away.
45¢ + ? = 100¢: count up from the price, one coin at a time, until you reach the dollar paid
Hand over coins until the count reaches 100¢
Each coin adds its length after the price. When the coins reach 100, the coins handed over are the change.
Worked example: Change Found by Counting Up
Question A bicycle costs $367. Mr Lim pays $500 for it. (a) How much must be added to $367 to reach $400? (b) How much change does Mr Lim get?
1.Start at 367 and jump to the next ten: 367 + 3 = 370.
Jump to the next ten: 367 + 3 = 370. 2.Jump to the next hundred: 370 + 30 = 400.
Jump to the next hundred: 370 + 30 = 400. 3.(a) The two jumps are 3 + 30 = 33, so $33 must be added to $367 to reach $400.
(a) 3 + 30 = 33, so $33 is added to reach $400. 4.Jump from 400 to 500. This jump is 100.
The jump from 400 to 500 is 100. 5.(b) Add all the jumps: 33 + 100 = 133, so Mr Lim gets $133 change. Check: 367 + 133 = 500.
(b) 33 + 100 = 133, so the change is $133.
Answer: (a) $33; (b) $133
Common mistakes
- Giving 500, the number where the jumps end, as the answer. The change is the sum of the jumps, 3 + 30 + 100.
- Jumping from 370 to 400 and calling the jump 130. From 370 to 400 is only 30, because 37 tens and 3 tens make 40 tens.
More adding and subtracting in your head problems, worked step by step →