https://file.aiquickdraw.com/mathbot/file/1727066296405wg87q7nk.jpg

Let's break down the problem step by step.

### Problem Restatement
You have quarters and dimes that total $2.80. Your friend suggests that the number of quarters is **8 more** than the number of dimes. We need to check if this claim is possible by modeling it with an equation, and verifying it.

Let **d** represent the number of dimes. Since each dime is worth $0.10 and each quarter is worth $0.25, we can write expressions for the total value of the dimes and the quarters.

### Step 1: Write the equation

1. **Number of Quarters**: If there are 8 more quarters than dimes, the number of quarters is represented by $ d + 8 $.
   
2. **Total Value of Dimes**: The value of the dimes is $ 0.10d $ (because each dime is worth $0.10).

3. **Total Value of Quarters**: The value of the quarters is $ 0.25(d + 8) $ (because each quarter is worth $0.25 and there are $ d + 8 $ quarters).

4. **Total Value of Both Coins**: The total value is given as $2.80, so the equation that models this situation is:

\[
0.10d + 0.25(d + 8) = 2.80
\]

### Step 2: Solve the Equation

Now, let's solve the equation to check if the values work.

\[
0.10d + 0.25(d + 8) = 2.80
\]
\[
0.10d + 0.25d + 2.00 = 2.80
\]
\[
0.35d + 2.00 = 2.80
\]
\[
0.35d = 0.80
\]
\[
d = \frac{0.80}{0.35} = \frac{80}{35} = 2.2857
\]

Since **d** is not a whole number, your friend is incorrect. The number of dimes must be a whole number, and this value doesn't satisfy that condition.

### Conclusion

Your friend is **not correct** because the number of dimes would have to be a fraction, which is not possible in this context (we can't have part of a coin).

---

Do you want further details or have any questions?

Here are five related questions:
1. How would the equation change if the total was $3.50 instead of $2.80?
2. What if the number of quarters was 5 more than the number of dimes—what would the equation be?
3. Could the equation have any solutions if we only had pennies and nickels instead?
4. How would you verify the solution if decimals weren't allowed in the final answer?
5. What if there were only nickels and quarters, how would the equation be different?

**Tip**: Always check that your solutions make sense in context—like making sure the number of coins is a whole number!

You have quarters and dimes that total $2.80. Your friend says it is possible that the number of quarters is 8 more than the number of dimes. Is your friend correct? Let d represent the number of dimes.

Learn how to solve a real-world problem involving the total value of quarters and dimes. We check if it is possible that the number of quarters is 8 more than the number of dimes, with a total of $2.80. Step-by-step algebraic solution provided. Suitable for Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation