Final answer:
The problem is solved by establishing the initial ratio of 5 to 10 dollar bills and determining the new amount of 5 dollar bills after converting 10 dollar bills. The final total value of all bills in the box is calculated to be $145.
Step-by-step explanation:
The question involves finding the total value of bills after a change in the ratio due to the conversion of 10 dollar bills into 5 dollar bills. Originally, the ratio of 5 dollar bills to 10 dollar bills was 5:7. After converting all 10 dollar bills to 5 dollar bills and putting them back into the box, the new ratio became 9:5.
Strategy
Let's denote the number of 5 dollar bills as 5x and the number of 10 dollar bills as 7x. Converting each 10 dollar bill into two 5 dollar bills gives us 14x more 5 dollar bills.
Solution
After conversion, the total number of 5 dollar bills is 5x + 14x = 19x. The new ratio is 9:5, meaning for 9 parts of 5 dollar bills, there are 5 parts of the initial 10 dollar bills left. The value of the bills can be calculated as:
Total value = (number of 5 dollar bills × $5) + (number of 10 dollar bills × $10).
Inserting our expressions in terms of x, we get:
Total value = (19x × $5) + (5x × $10)
= 95x + 50x
= 145x dollars.
To find the value of x, we use the new ratio:
19x / (5x) = 9/5 →
x = 1.
Therefore, the total value of the bills is 145 × 1 = $145.