Let's start by using variables to represent the unknown quantities in the problem. Let:
- x be the number of fifty-cent coins in the box
- y be the number of twenty-cent coins in the box
From the first sentence of the problem, we know that:
y = (1/3)x
We can use this equation to substitute y in terms of x in the rest of the problem.
Next, we're told that 8 fifty-cent coins were exchanged for twenty-cent coins. This means that:
- 8 fifty-cent coins were taken out of the box
- 8 twenty-cent coins were put into the box
After this exchange, the number of fifty-cent coins became 4/7 the number of twenty-cent coins. This means that:
x - 8 = (4/7)(y + 8)
We can substitute y in terms of x from the first equation to get:
x - 8 = (4/7)((1/3)x + 8)
Simplifying this equation, we get:
x - 8 = (4/21)x + (32/7)
Multiplying both sides by 21 to eliminate fractions, we get:
21x - 168 = 4x + 96
Solving for x, we get:
x = 18
Now we can use this value of x to find y:
y = (1/3)x = 6
This means there were originally 18 fifty-cent coins and 6 twenty-cent coins in the box, for a total of $9.60:
18 x $0.50 = $9.00
6 x $0.20 = $1.20
Therefore, the answer is $9.60.