Question

Henry has a collection of dimes and quarters totaling $47.50. The number of quarters is 10 more than twice the number of dimes. If d represents the number of dimes and q represents the number of quarters, select the system of equations described by the scenario.

Answer 1

To set up the system of equations based on the scenario provided, let's use \( d \) to represent the number of dimes and \( q \) to represent the number of quarters.

1. **Total Value Equation**:
The total value of the dimes and quarters equals $47.50. Since each dime is worth $0.10, the value from the dimes is \( 0.10d \). Each quarter is worth $0.25, so the value from the quarters is \( 0.25q \). Therefore, the first equation is:
\[
0.10d + 0.25q = 47.50
\]

2. **Relationship Between Quarters and Dimes**:
According to the problem, the number of quarters is 10 more than twice the number of dimes. Mathematically, we can express this relationship as:
\[
q = 2d + 10
\]

Now we have a system of equations:

1. \( 0.10d + 0.25q = 47.50 \)
2. \( q = 2d + 10 \)

You can also multiply the first equation by 100 to eliminate the decimals if preferred:

1. \( 10d + 25q = 4750 \)
2. \( q = 2d + 10 \)

So the final system of equations is:
\[
\begin{cases}
10d + 25q = 4750 \\
q = 2d + 10
\end{cases}
\]