A man has 13 coins in his pocket, all of which are dimes and quarters. If the total value of his change is 200 and $.50 how many dimes and how many quarters does he have?

To solve this problem, we need to set up a system of equations. Let $ d $ represent the number of dimes, and $ q $ represent the number of quarters.

We have two pieces of information:

1. The total number of coins is 13.
2. The total value of the coins is $2.50.

First, we convert the total value to cents to avoid dealing with decimals. Therefore, $2.50 is equivalent to 250 cents.

Given:
1. $ d + q = 13 $
2. $ 10d + 25q = 250 $

Now we can solve this system of equations.

### Step 1: Solve for one variable
Let's solve the first equation for $ d $:

\[ d = 13 - q \]

### Step 2: Substitute into the second equation
Substitute $ d = 13 - q $ into the second equation:

\[ 10(13 - q) + 25q = 250 \]

### Step 3: Simplify and solve for $ q $

\[ 130 - 10q + 25q = 250 \]

\[ 130 + 15q = 250 \]

\[ 15q = 250 - 130 \]

\[ 15q = 120 \]

\[ q = \frac{120}{15} \]

\[ q = 8 \]

### Step 4: Find $ d $
Now that we have $ q = 8 $, we can find $ d $:

\[ d = 13 - q \]

\[ d = 13 - 8 \]

\[ d = 5 \]

So, the man has 5 dimes and 8 quarters.

### Verification
Let's verify the solution by checking the total value:

- Value of 5 dimes: $ 5 \times 10 = 50 $ cents
- Value of 8 quarters: $ 8 \times 25 = 200 $ cents

Total value: $ 50 + 200 = 250 $ cents or $2.50.

The solution is verified.

### Summary
The man has 5 dimes and 8 quarters.

Would you like details or have any questions?

Here are some related questions for further practice:
1. What if the total value of the coins was $3.00, and he still had 13 coins?
2. How would the system change if he had 10 coins instead of 13, but the total value remained $2.50?
3. If the number of dimes was doubled and the total number of coins was 20, what would be the new total value?
4. Suppose the total number of coins was 15, and their value was $3.50. How many dimes and quarters would he have?
5. How would you approach the problem if instead of dimes and quarters, he had nickels and quarters?
6. What if he had 13 coins consisting of dimes and nickels, and their total value was $1.30?
7. How would you adjust the equations if the coins included nickels, dimes, and quarters, and the total value was $2.50?
8. How does changing the coin values (e.g., from cents to another currency) affect the solution?

**Tip:** When solving word problems involving coins, always convert the total value into the smallest unit (like cents) to simplify calculations.

Learn how to solve a word problem involving dimes and quarters. Understand the concept of systems of equations and apply it to find the number of dimes and quarters when given the total number of coins and their value. Suitable for middle school students in Grades 6-8.

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