Problem 169 - Elementary Algebra

Question

Harrison has \(\$ 9.30\) in his coin collection, all in pennies and dimes. The number of dimes is three times the number of pennies. How many coins of each type does he have?

Step-by-Step Solution

Answer

30 pennies and 90 dimes

Step 1 - Define Variables

Let the number of pennies be represented by the variable \( p \). Given that the number of dimes is three times the number of pennies, let the number of dimes be represented by the variable \( 3p \).

Step 2 - Set Up the Equation

Each penny is worth \( \$0.01 \) and each dime is worth \( \$0.10 \). Since Harrison has \( \$9.30 \), the total value equation can be written as: \[ 0.01p + 0.10(3p) = 9.30 \]

Step 3 - Simplify the Equation

Simplify the equation by combining like terms: \[ 0.01p + 0.30p = 9.30 \]Combine the coefficients of \( p \): \[ 0.31p = 9.30 \]

Step 4 - Solve for \( p \)

Solve for \( p \) by dividing both sides of the equation by \( 0.31 \): \[ p = \frac{9.30}{0.31} \]Calculate the value: \[ p = 30 \]

Step 5 - Find the Number of Dimes

Since the number of dimes is three times the number of pennies: \[ 3p = 3 \times 30 = 90 \]

Key Concepts

Defining Variables

When faced with an algebra word problem, the first step is to understand what needs to be found and to represent those unknowns using variables. In our problem, we need to find the number of pennies and dimes Harrison has. Let's start with:

• Let the number of pennies be represented by the variable \( p \).
• The problem states that the number of dimes is three times the number of pennies. Therefore, we represent the number of dimes as \( 3p \).

Defining variables is crucial because it transforms the problem into a format that can be solved using algebra.

Setting Up Equations

After defining the variables, the next step is to set up an equation based on the information given. Each penny is worth \( \$0.01 \) and each dime is worth \( \$0.10 \). The total value of Harrison's coins is \( \$9.30 \). Using this information, we set up the equation:

\[ 0.01p + 0.10(3p) = 9.30 \]
This equation represents the total value of all pennies and dimes combined. Setting up accurate equations is essential because they form the basis for finding the solution to the problem.

Solving Linear Equations

Once we have our equation set up, the next step is to simplify and solve it. Here's the equation simplified:

• First, expand the equation: \( 0.01p + 0.30p = 9.30 \)
• Combine like terms: \( 0.31p = 9.30 \)
• To solve for \( p \), divide both sides by \( 0.31 \): \( p = \frac{9.30}{0.31} = 30 \)

So, Harrison has 30 pennies. Solving the equation correctly is critical as it directly provides us with the number of pennies.

Mathematical Reasoning

Mathematical reasoning ties together the entire process of solving the problem. Here, we use logical steps to find the final solution.

First, we identified the relationship between the number of pennies and dimes. Then, using this relationship and the total monetary value, we set up and solved a linear equation.

After finding the number of pennies (\( p = 30 \)), we determine the number of dimes, which is three times the number of pennies: \( 3p = 3 \times 30 = 90 \).

So, Harrison has:

• 30 pennies
• 90 dimes

Good mathematical reasoning ensures each step is logical and consistent, which helps in solving complex algebraic word problems with confidence.