Give you a clue: if each of them spent half their coins, then as a whole, half of the coins are spent; it does not matter whether one person has 250 coins and the other has 200, or one person has 100 coins and the other has 300, or any other combination

So, after the usage of the $1 coins, they will have a total of $225 left

This becomes a case of “in the end, they had $225 left, with Sharon having $51 more than Alice”

Do you know what to do after this?

Alice in the end: 1 unit
Sharon in the end: 1 unit + $51

2 units + $51 = $225
2 units = $174
1 unit = $87

So, in the end, Alice had $87 (ie 87 pieces of $1 coins) while Sharon had $138 (ie 138 pieces of $1 coins)

Sharon had double of this, or 276 pieces of $1 coins, at first

This idea only works when both of them spent the same fraction of money

If Sharon spent 2/3 of her money and Alice spent 1/4 of hers instead, this idea would not work