there are 50 coins that total $9.20. The coins are only dimes and quarters. How many quarters are there 

Let the number of quarters be denoted as $ q $ and the number of dimes as $ d $.

We know:
1. There are 50 coins in total, so:
   \[
   q + d = 50
   \]
2. The total value of the coins is $9.20, which is 920 cents. Since quarters are worth 25 cents and dimes are worth 10 cents, we have:
   \[
   25q + 10d = 920
   \]

We now have a system of two equations:
\[
\begin{aligned}
q + d &= 50 \quad \text{(1)} \\
25q + 10d &= 920 \quad \text{(2)}
\end{aligned}
\]

### Solving the system of equations:
From equation (1), solve for $ d $:
\[
d = 50 - q
\]

Substitute this into equation (2):
\[
25q + 10(50 - q) = 920
\]

Simplify and solve for $ q $:
\[
25q + 500 - 10q = 920
\]
\[
15q + 500 = 920
\]
\[
15q = 920 - 500
\]
\[
15q = 420
\]
\[
q = \frac{420}{15} = 28
\]

Thus, there are **28 quarters**.

### Checking the solution:
If there are 28 quarters, then the number of dimes is:
\[
d = 50 - 28 = 22
\]
The total value is:
\[
25(28) + 10(22) = 700 + 220 = 920 \, \text{cents} = 9.20 \, \text{dollars}
\]
The solution is correct.

Would you like more details, or do you have any further questions?

Here are 5 related questions:
1. What if there were 60 coins and the same total value of $9.20?
2. How would the solution change if the total value was $10.00 with 50 coins?
3. What if the number of coins was unknown but the total value was still $9.20?
4. How can you solve this problem using matrices or graphically?
5. How would you approach this if pennies were included in the mix?

**Tip**: When working with coin problems, setting up a system of equations based on the number of coins and their total value can simplify finding the solution.

There are 50 coins that total $9.20. The coins are only dimes and quarters. How many quarters are there?

Learn how to determine the number of quarters in a set of 50 coins that total $9.20, using algebra and systems of equations. This problem involves solving for the number of dimes and quarters based on their value and total number of coins. Suitable for Grades 6-8.

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