Math Problem Statement

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?

Solution

To solve this problem, we need to set up a system of equations. Let dd represent the number of dimes, and qq 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=13d + q = 13
  2. 10d+25q=25010d + 25q = 250

Now we can solve this system of equations.

Step 1: Solve for one variable

Let's solve the first equation for dd:

d=13qd = 13 - q

Step 2: Substitute into the second equation

Substitute d=13qd = 13 - q into the second equation:

10(13q)+25q=25010(13 - q) + 25q = 250

Step 3: Simplify and solve for qq

13010q+25q=250130 - 10q + 25q = 250

130+15q=250130 + 15q = 250

15q=25013015q = 250 - 130

15q=12015q = 120

q=12015q = \frac{120}{15}

q=8q = 8

Step 4: Find dd

Now that we have q=8q = 8, we can find dd:

d=13qd = 13 - q

d=138d = 13 - 8

d=5d = 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: 5×10=505 \times 10 = 50 cents
  • Value of 8 quarters: 8×25=2008 \times 25 = 200 cents

Total value: 50+200=25050 + 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.

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Math Problem Analysis

Mathematical Concepts

Systems of Equations
Word Problems
Coin Problems

Formulas

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Theorems

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Suitable Grade Level

Grades 6-8