Let's denote the number of dimes as \( d \) and the number of quarters as \( q \).
We have the following two equations from the problem statement:
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The total number of coins: \[ d + q = 250 \]
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The total value of the coins (in cents): \[ 10d + 25q = 3925 \] (Note: The value in dollars is converted to cents for easier calculations.)
Now we can solve these equations step-by-step.
From the first equation, we can express \( q \) in terms of \( d \): \[ q = 250 - d \]
Next, we substitute \( q \) into the second equation: \[ 10d + 25(250 - d) = 3925 \]
Expanding the equation: \[ 10d + 6250 - 25d = 3925 \]
Combining like terms: \[ -15d + 6250 = 3925 \]
Now, let's isolate \( d \): \[ -15d = 3925 - 6250 \] \[ -15d = -2325 \]
Dividing both sides by -15: \[ d = \frac{2325}{15} = 155 \]
Now that we have \( d \), we can find \( q \): \[ q = 250 - d = 250 - 155 = 95 \]
Thus, the number of dimes is \( d = 155 \) and the number of quarters is \( q = 95 \).
Let's summarize:
- Number of dimes: 155
- Number of quarters: 95
We can double-check the values:
Calculating the total value:
- Value from dimes: \( 155 \times 10 = 1550 \) cents
- Value from quarters: \( 95 \times 25 = 2375 \) cents
- Total value: \( 1550 + 2375 = 3925 \) cents (which corresponds to $39.25)
Everything checks out correctly. Therefore, the solution is:
- Dimes: 155
- Quarters: 95
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