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 represent the number of dimes, and represent the number of quarters.
We have two pieces of information:
- The total number of coins is 13.
- 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:
Now we can solve this system of equations.
Step 1: Solve for one variable
Let's solve the first equation for :
Step 2: Substitute into the second equation
Substitute into the second equation:
Step 3: Simplify and solve for
Step 4: Find
Now that we have , we can find :
So, the man has 5 dimes and 8 quarters.
Verification
Let's verify the solution by checking the total value:
- Value of 5 dimes: cents
- Value of 8 quarters: cents
Total value: 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:
- What if the total value of the coins was $3.00, and he still had 13 coins?
- How would the system change if he had 10 coins instead of 13, but the total value remained $2.50?
- If the number of dimes was doubled and the total number of coins was 20, what would be the new total value?
- Suppose the total number of coins was 15, and their value was $3.50. How many dimes and quarters would he have?
- How would you approach the problem if instead of dimes and quarters, he had nickels and quarters?
- What if he had 13 coins consisting of dimes and nickels, and their total value was $1.30?
- How would you adjust the equations if the coins included nickels, dimes, and quarters, and the total value was $2.50?
- 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
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