Question

solving a mixture problem
number value total value
pennies p 0.01 0.01p
nickels 22 - p 0.05 0.05(22 - p)
totals 22 0.54
ari has a total of 22 coins consisting of pennies and nickels. the total value of the coins is $0.54. how many of each type of coin does ari have?
pennies and nickels

Explanation:

Step1: Set up the equation

The total value of pennies is \(0.01p\) and the total value of nickels is \(0.05(22 - p)\). The sum of these two should equal the total value of the coins, which is \(0.54\). So the equation is \(0.01p+0.05(22 - p)=0.54\).

Step2: Expand the equation

First, expand \(0.05(22 - p)\) using the distributive property. \(0.05\times22 = 1.1\) and \(0.05\times(-p)=- 0.05p\). So the equation becomes \(0.01p + 1.1-0.05p=0.54\).

Step3: Combine like terms

Combine the \(p\) terms: \(0.01p-0.05p=-0.04p\). So the equation is now \(-0.04p + 1.1 = 0.54\).

Step4: Solve for \(p\)

Subtract \(1.1\) from both sides: \(-0.04p=0.54 - 1.1\). Calculate \(0.54 - 1.1=-0.56\). Then divide both sides by \(-0.04\): \(p=\frac{-0.56}{-0.04}\). \(p = 14\).

Step5: Find the number of nickels

The number of nickels is \(22 - p\). Substitute \(p = 14\) into this: \(22-14 = 8\).

Answer:

14 pennies and 8 nickels