Question

1. there were 24 pennies, nickels, and quarters in all, and their total value was $1.60. how many of each kind of coin were there if the number of nickels equaled the number of pennies?

Explanation:

Step1: Define Variables

Let \( p \) be the number of pennies, \( n \) be the number of nickels, and \( q \) be the number of quarters. We know that \( p = n \) (number of nickels equals number of pennies), and the total number of coins is \( p + n + q = 24 \). The total value is \( 0.01p + 0.05n + 0.25q = 1.60 \).

Step2: Substitute \( n \) with \( p \) in Total Coins Equation

Since \( n = p \), substitute into \( p + n + q = 24 \):
\( p + p + q = 24 \)
Simplify: \( 2p + q = 24 \)
We can express \( q \) as \( q = 24 - 2p \).

Step3: Substitute \( n \) and \( q \) in Total Value Equation

Substitute \( n = p \) and \( q = 24 - 2p \) into \( 0.01p + 0.05n + 0.25q = 1.60 \):
\( 0.01p + 0.05p + 0.25(24 - 2p) = 1.60 \)

Step4: Simplify and Solve for \( p \)

Combine like terms:
\( 0.06p + 6 - 0.5p = 1.60 \)
\( -0.44p + 6 = 1.60 \)
Subtract 6 from both sides:
\( -0.44p = 1.60 - 6 \)
\( -0.44p = -4.4 \)
Divide by \(-0.44\):
\( p = \frac{-4.4}{-0.44} = 10 \)

Step5: Find \( n \) and \( q \)

Since \( n = p \), \( n = 10 \).
From \( q = 24 - 2p \):
\( q = 24 - 2(10) = 4 \)

Answer:

There are 10 pennies, 10 nickels, and 4 quarters.