Question

Rob has 40 coins, all dimes and quarters, worth $7.60. How many dimes and how many quarters does he have?

Answer 1

Let \( d \) represent the number of dimes and \( q \) represent the number of quarters. We can set up the following system of equations based on the information provided:

1. The total number of coins is 40:
\[
d + q = 40
\]

2. The total value of the coins is $7.60. Since dimes are worth $0.10 and quarters are worth $0.25, we can express this as:
\[
0.10d + 0.25q = 7.60
\]

To eliminate the decimals in the second equation, we can multiply the entire equation by 100:
\[
10d + 25q = 760
\]

Now we have the following system of equations:
1. \( d + q = 40 \)
2. \( 10d + 25q = 760 \)

Next, we can solve the first equation for \( d \):
\[
d = 40 - q
\]

Now we can substitute \( d \) in the second equation:
\[
10(40 - q) + 25q = 760
\]
\[
400 - 10q + 25q = 760
\]
\[
400 + 15q = 760
\]
\[
15q = 760 - 400
\]
\[
15q = 360
\]
\[
q = \frac{360}{15} = 24
\]

Now that we have \( q \), we can find \( d \):
\[
d = 40 - q = 40 - 24 = 16
\]

Thus, Rob has:
- \( 16 \) dimes
- \( 24 \) quarters

To verify:
- Total number of coins: \( 16 + 24 = 40 \) (correct)
- Total value: \( 16 \times 0.10 + 24 \times 0.25 = 1.60 + 6.00 = 7.60 \) (correct)

The solution checks out. Hence, the final answer is:
\[
\boxed{16} \text{ dimes and } \boxed{24} \text{ quarters.}
\]