Box A: 50-cent coins only
Box B: 20-cent coins only

Box B has 60 more coins than Box A
Box A has $4.50 more than Box B

If Box B has 60 coins fewer than currently, so that Box B and Box A have the same number of coins, then the value of the coins in Box B drops by 60 x 20c = $12.

In this scenario, Box A has $16.50 more than Box B when both boxes have the same number of coins.

For each 50c coin versus each 20c coin considered, the value is 30c in difference.

For a total of $16.50 difference, or 1650 cents, number of sets of coins = 1650/30 = 55.

This means that in this scenario, there are 55 coins in Box A and 55 coins in Box B.

There must have been 55 coins in Box A and (55 + 60) = 115 coins in Box B at first.

Thank you very much for your detailed working!