Notice that the total value of the coins didn't change.

20¢ x 30 = 600¢
600¢ ÷ 50¢ = 12

So 30 twenty-cent coins were exchanged for 12 fifty-cent coins.

Ratio of fifty cent to twenty cent at
first = 4 : 5

Multiply this by the value of the coins,

4 x 50¢ : 5 x 20¢
= 200¢ : 100¢

Total = 300¢

Ratio at the end = 7 : 5

Multiply this by the value of the coins,

7 x 50¢ : 5 x 20¢
= 350 ¢ : 100 ¢

Total = 450¢

But we know that total value at the start and at the end is the same. So we have to make the total the same.

300¢ x 3 = 900¢
450¢ x 2 = 900¢

So we should multiply the starting ratio by
3, and the end ratio by 2.

Starting ratio x 3 → 4 x 3 : 5 x 3
= 12 : 15

End ratio x 2→ 7 x 2 : 5 x 2
= 14 : 10

Ratio of difference = 2 : 5

Remember that 30 twenty-cent coins were exchanged for 12 fifty-cent coins.

So, difference between starting and end number of 20¢ coins is 30. Difference between starting and end number of 50¢ coins is 12.

We need to change the ratio of difference.
So multiply by 6.

2 : 5 → x6 → 12 : 30

So this means our starting ratio and end ratio should also be multiplied by 6.

Starting → 12 : 15 → 72 : 90

End → 14 : 10 → 84 : 60