A box contained some twenty-cent, fifty-cent and one-dollar coins. $24\%$ of the coins are twenty-cent coins. The ratio of the number of fifty-cent coins to the number of one-dollar coins is 7 : 12. When $62.5\%$ of the twenty-cent coins were removed, the total number of coins decreased by 15%. In the end there were 255 coins left.

1. Determine the percentage of total coins remaining

Since the total number of coins decreased by $15\%$, the percentage of coins left is:
$100\% - 15\% = 85\%$

2. Calculate the number of coins removed

The $85\%$ remaining corresponds to $255$ coins.
The number of coins removed is the $15\%$ decrease.
Number of coins removed = $\frac{15}{85} \times 255 = 45$
Since only twenty-cent coins were removed, $45$ twenty-cent coins were removed.

1. Find the percentage of the total coins that are one-dollar coins.

Percentage of fifty-cent and one-dollar coins $= 100\% - 24\% = 76\%$

Ratio of fifty-cent to one-dollar coins is $7 : 12$. Total parts $= 7 + 12 = 19$.

Percentage of one-dollar coins $= \frac{12}{19} \times 76\% = 48\%$

2. Find the value of 1% of the total coins using the final number of coins.

Percentage of coins left $= 100\% - 15\% = 85\%$

$85\%$ of total coins $= 255$
$1\%$ of total coins $= \frac{255}{85} = 3$

3. Calculate the number of one-dollar coins.

Number of one-dollar coins $= 48 \times 3 = 144$