The Gap-and-Difference Method

Some problems give you two plans for the same situation: one plan leaves a bit extra, and the other plan falls a bit short. Hidden inside those two gaps is the answer — usually a number of items you could not see at first. This is the gap-and-difference method (also called excess and shortage), and it is the grown-up replacement for slow guess-and-check.

Parents: let your child slide the count and watch the two plans meet before revealing. The "Method tip" boxes name the working a marker rewards — total gap divided by the gap per item.

By the end you'll be able to spot an excess-and-shortage problem, add the two gaps, and divide to find the hidden number. Let's close the gap.

Two plans, two gaps

The set-up always has the same shape. You have a fixed amount (say, some money). Plan one shares it out at one rate and there is some left over — that is the excess. Plan two shares it at a bigger rate and there is not enough — that is the shortage.

The two plans describe the same hidden amount, so the only thing that changes between them is the rate per item. That is what lets you find the number of items.

Add the gaps, then divide

Try the classic. Mrs Lim fills red packets. If she puts $30 in each, she has $20 left over. If she puts $40 in each, she is $30 short. How many red packets are there? Slide the number of packets until both plans point to the same total amount of money.

🔬

Red packets: close the gap

🤔 Predict first: Going from $30 to $40 in each packet, what happens to the money needed?

It goes upIt goes down

Here is the reasoning. Between the two plans, the total gap is the $20 you had spare plus the $30 you were missing — that is $50. Each packet changed by $40 − $30 = $10. So the number of packets is the total gap divided by the gap per packet: 50 ÷ 10 = 5 packets. (Her money was 30 × 5 + 20 = $170.)

Better than guess-and-check

You could solve these by guess-and-check, and at P3 that is fine. But gap-and-difference gets the answer in one division instead of many guesses. It is part of the same family as the supposition method, where you assume one extreme (for example, "suppose every vehicle is a motorcycle") and reason from how far off the total is.

Watch out — these are easily mixed up

Quick recap

🎯 Mastery check

Answer all 6 — your progress is saved on this device.

1. One plan leaves $15 over and another plan is $25 short. What is the total gap?

   $10$40$25

2. Putting $20 in each box leaves money over; putting $26 in each falls short. What is the gap per box?

   $6$46$13

3. The total gap is $60 and the gap per item is $12. How many items are there?

   57248

4. $10 in each packet leaves $8 over; $14 in each packet is $12 short. How many packets?

   546

5. Why do you ADD the excess and the shortage instead of subtracting?

   Because adding is easierBecause together they measure the full change between the two plansBecause the answer must be positive

6. Giving each pupil 3 stickers leaves 9 over; giving each pupil 4 leaves 6 short. How many pupils, and what is the first step?

   15, after total gap 15 ÷ per pupil 13, by dividing 9 by 39, the leftover

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