An, Bình, and 133 pebbles

A guide, not a test. Nothing here is scored and nothing is sent anywhere.

You got this one right. 7 parts, one part is 19, so An 57 and Bình 76 — and you wrote the lines in the right order too.

So this page is not a correction. A right answer proves you can run the method; it does not yet prove you could bend it. That is what the rest of this is for — first why the trick is allowed at all, then what you do when the question hands you a difference instead of a total.

1Play the game and watch

Nobody handed them 133 pebbles and asked them to share it out. They took them, a bit at a time. Every time An grabs 3, Bình grabs 4. Do that once and call it a round.

Slide, and watch the two piles grow.

And once you know it was 19 rounds, both answers fall out without any more thinking: An took 3 nineteen times, so 3 × 19 = 57. Bình took 4 nineteen times, so 4 × 19 = 76.

This is worth noticing, because it is the reason the whole method is allowed. A "part" is not a made-up thing you draw to make the sum work. In this question a part is a real round of the game.

2Your drawing is the same thing, squashed

You drew An as 3 boxes and Bình as 4 boxes, all the same size, with 133 down the side. That picture is 19 rounds stacked on top of each other — each box is one pebble per round, nineteen times over.

Drag the box size and watch the total. There is exactly one setting that makes it 133.

Sliding until it fits is fine for seeing it. It is not how you answer it, because you would be guessing. Going the other way — from 133 straight back to the box — is one division: 133 ÷ 7 = 19.

3The three lines you wrote

That order is not a style choice. Each line only uses things the line above has already found, which is why you can always write the first one even on a question you have not worked out yet — and writing it is often what shows you the rest.

4Check it two different ways

One check can agree with a mistake, because the mistake is already in your head. Two checks that look at different things cannot both be fooled by the same slip.

Both agree. Now you are allowed to believe it.

The second check is the one people skip, and it is the one that catches the worst mistake — giving the right total to the wrong two people. 66 + 67 also makes 133.

5Three ways to get it wrong

All three of these have been written by someone who understood the question perfectly well. Tap each to see what went wrong.

6Now change the question

This is the part that matters. The numbers in a real paper will not be 3, 4 and 133 — and quite often you will not be given the total at all. You will be told how many more one of them has.

Move anything. The three lines rewrite themselves.

Try the 3 : 4, total 100 button. It refuses, and it is right to: 100 is not made of 7s, so there is no whole-pebble answer. Which tells you something about the paper — 133 was chosen. When your division comes out with a remainder on a question like this, do not round it. Go back and check you read the ratio the right way round.

7Your turn — four quick ones

No score, no timer. A wrong pick tells you why and lets you try again.

8The lines to write every time

Step 2 is the whole question in disguise. Everything after it is arithmetic you can already do — and everything before it is just reading carefully, which on a paper like this is worth more marks than being fast.