Let \( p \) represent the number of pennies that Ethan has, and let \( n \) represent the number of nickels. According to the problem, we have the following information:
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The total value of the pennies and nickels is $0.75. Since each penny is worth $0.01 and each nickel is worth $0.05, we can express this as: \[ 0.01p + 0.05n = 0.75 \]
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Ethan has 9 more nickels than pennies, which can be written as: \[ n = p + 9 \]
Now, we can substitute the second equation into the first equation. First, let's express the first equation in terms of \( p \) only:
By substituting \( n \): \[ 0.01p + 0.05(p + 9) = 0.75 \]
Now, distribute \( 0.05 \) in the equation: \[ 0.01p + 0.05p + 0.45 = 0.75 \]
Combine the \( p \) terms: \[ 0.06p + 0.45 = 0.75 \]
Next, we can isolate \( p \) by subtracting \( 0.45 \) from both sides: \[ 0.06p = 0.75 - 0.45 \] \[ 0.06p = 0.30 \]
Now, divide both sides by \( 0.06 \) to find \( p \): \[ p = \frac{0.30}{0.06} = 5 \]
Now that we have the number of pennies, we can use the second equation to find \( n \): \[ n = p + 9 = 5 + 9 = 14 \]
So, the solution is:
- Number of pennies = \( 5 \)
- Number of nickels = \( 14 \)
To confirm, let's check the total value:
- Value from pennies: \( 5 \times 0.01 = 0.05 \)
- Value from nickels: \( 14 \times 0.05 = 0.70 \)
- Total value: \( 0.05 + 0.70 = 0.75 \)
The calculations are correct. Therefore, Ethan has:
\[ \text{Number of Pennies} = 5, \quad \text{Number of Nickels} = 14 \]
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