Fun with numbers

The clever bit is...

1. Call the four numbers a, b, c and d.

2. One of the answers is a x c (2) and one of them is b x d (6). Multiply these up is a x b x c x d (12).

3. One of the answers is a x d (3) and one of them is b x c (4). Multiply these up is also a x b c x d (12).

4. You find these by trial and error, which doesn't take too long.

5. The multiple of the other two pairs, c x d (5) times a x b (?) must also = 12. So a x b must be 2.4
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Just to check whether I had understood it all properly, I ground out the answers:

(also for the benefit of Shivam Shukla)

6. We know that b x c = 4 and a x c = 2, so b = 2 x a.
a x b = 2.4 (from 5. above).
So a x 2a = 2.4.
So 2a^2 = 2.4
So a^2 = 1.2
So a = sq root of 1.2, or sq root of 6/5.

7. To find b:
b = 2a (from 6. above)
To multiply the sq root of a fraction (sq root 6/5) by a whole number (2), you:
- multiply the top number (6) by the number (2) squared (4) = 24.
- leave the bottom of the fraction (5) as it is.
So b = sq root of 24/5.

8. To find c:
a x b = 2.4 and a x c = 2.
So c = 2/2.4 x b = 5/6 x b
To multiply the sq root of a fraction (sq root of 24/5) by a fraction (5/6) you:
- multiply the top number (24) by the top number of the fraction squared (25) = 600.
- multiply the bottom number (5) by the bottom number of the fraction (6) squared (36) = 180
- 600/180 simplifies to 10/3
So c = sq root of 10/3.

9. You find d the same way you found c.
It is c x 3/2.
c = sq root of 10/3
d = sq root of 90/12 = sq root of 15/2.