The bag contains "some" 20-cent and "some" 50-cent coins. I assume that some means more than 1, so I start assuming the bag had 2 50-cent coins. Since there were 14 fewer 50-cent coins than 20-cent coins, for any guess of the number of 50-cent coins, you must use 14 more 20-cent coins.
1st guess: 2 50-cent coins + 16 20-cent coins
Value: 2 * $0.5 + 16 * $0.2 = $4.20
Since the difference in the numbers of coins is always 14, each time we add one 50-cent coin and 1 20-cent coin, the value goes up by $0.70.
$27.30 - $4.20 = $23.10
Our first guess is low in value by $23.10. How many additions of $0.70 (one coin of each) will it take to get the full value of $27.30?
$23.10/$0.70 = 33
By adding 33 pairs of coins to the 2 50-cent coins and 16 20-cent coins, the value will be $27.30.
Number of 50-cent coins = 2 + 33 = 35
Check: If there are 35 50-cent coins, there are 35 + 14 = 49 20-cent coins.
Value: 35 * $0.5 + 49 * $0.2 = $17.50 + $9.80 = $27.30
35 50-cent coins plus 49 20-cent coins do add up to a value of $27.30, so answer of 35 50-cent coins is correct.
