Ben has twice as many marbles as Cara. If Ben gives Cara 6 marbles, they will have the same number. How many marbles does Ben have at first?
You left this one blank, and it is a fair thing to leave blank — it hides a trick that almost everybody gets wrong the first time. Once you have seen the trick it is one of the easiest kinds of question in the paper.
Let's find it. Take your time; there is no clock on this page.
Two jars. One has 10 marbles, the other has 4. Move marbles from the full one to the empty-ish one and watch the gap between them.
| You moved | 0 |
|---|---|
| The gap shrank by | 0 |
Slide it all the way and look at those two numbers side by side. Move 3, and the gap does not shrink by 3 — it shrinks by 6.
Moving n across closes a gap of 2n.
One side goes down by n and the other goes up by n. It counts twice.
That is the whole trick. Everything below is just using it.
We do not know how many marbles anyone has yet — but we know the shape. Cara is one lot. Ben is twice as many, so Ben is two of the same lot. Call one lot a unit.
Look at the picture, not the words: Ben's bar sticks out past Cara's by one unit. That sticking-out bit is the gap.
Tap through it one line at a time.
Ben starts with 24 and Cara with 12. Move 6 across and see them land level.
Notice it does not stop at equal — push past 6 and Cara goes ahead. Six is the exact point where the gap of 12 runs out, because 6 + 6 = 12.
No score, no timer. If you pick a wrong one it will just tell you why and let you try again.
Ann has 7 more sweets than Bob. Ann gives Bob 3 sweets. How many more does Ann have than Bob now?
Sam has twice as many cards as Tim. If Sam gives Tim 5 cards, they will have the same number. How many cards did Sam have at first?
1. Draw the bars in units. The smaller person is 1 unit.
2. The gap is however many units stick out. Moving n across closes 2n — so if they end up equal, gap = 2n.
3. Work out one unit, then read the question again and answer the thing it actually asked.
That last line is worth as much as the other two. On your first drill you worked out 3/8 of 40 = 15 perfectly and then answered 15 — but the question asked how many were left, which was 25. The maths was right; the last read was the miss.