Problem 61 - Algebra for College Students

Question

Use an algebraic approach to solve each problem. Suppose that Maria has 150 coins consisting of pennies, nickels, and dimes. The number of nickels she has is 10 less than twice the number of pennies; the number of dimes she has is 20 less than three times the number of pennies. How many coins of each kind does she have?

Step-by-Step Solution

Answer

Maria has 30 pennies, 50 nickels, and 70 dimes.

Step 1: Define the Variables

Let \( p \) represent the number of pennies, \( n \) represent the number of nickels, and \( d \) represent the number of dimes. We know the total number of coins is \( 150 \), thus our first equation is \( p + n + d = 150 \).

Step 2: Express Nickels in Terms of Pennies

According to the problem, the number of nickels is 10 less than twice the number of pennies. This gives us the equation \( n = 2p - 10 \).

Step 3: Express Dimes in Terms of Pennies

The number of dimes is 20 less than three times the number of pennies. Therefore, \( d = 3p - 20 \).

Step 4: Substitute the Expressions into the First Equation

Substitute \( n \) and \( d \) from steps 2 and 3 into the first equation: \( p + (2p - 10) + (3p - 20) = 150 \).

Step 5: Simplify the Equation

Combine like terms to simplify: \( p + 2p - 10 + 3p - 20 = 150 \) becomes \( 6p - 30 = 150 \).

Step 6: Solve for Pennies

Add 30 to both sides of the equation: \( 6p = 180 \). Then divide by 6: \( p = 30 \).

Step 7: Solve for Nickels

Substitute \( p = 30 \) back into the equation for nickels: \( n = 2(30) - 10 = 60 - 10 = 50 \).

Step 8: Solve for Dimes

Substitute \( p = 30 \) back into the equation for dimes: \( d = 3(30) - 20 = 90 - 20 = 70 \).

Step 9: Verify the Solution

Add all three to check the total number of coins: \( p + n + d = 30 + 50 + 70 = 150 \), which matches the given condition.

Key Concepts

Linear Equations

A linear equation is an essential tool in algebra used to describe relationships between different quantities. In our problem, we connect the total number of pennies, nickels, and dimes that Maria has. A linear equation often takes the form of a simple equation like:

• \(p + n + d = 150\)

This equation shows the sum of all the coins, totaling 150. Here, each letter represents the number of coins: \(p\) for pennies, \(n\) for nickels, and \(d\) for dimes. Breaking down this equation helps us solve the problem systematically.

When solving linear equations, we may rearrange terms to isolate variables or substitute expressions for others, just as we substitute for \(n\) and \(d\) using expressions derived from the problem conditions. This technique, called substitution, simplifies solving multiple unknowns by reducing the number of variables in the equation.

Variables in Algebra

Variables are symbols used to represent unknown numbers or quantities in mathematical expressions and equations. In this problem, the letters \(p\), \(n\), and \(d\) act as placeholders for unknown values:

• \(p\) represents the number of pennies
• \(n\) stands for the number of nickels
• \(d\) signifies the number of dimes

Understanding and defining these variables is the starting point for solving the problem. The relationships given in the problem allow us to express \(n\) and \(d\) in terms of \(p\), which is convenient and simplifies the solution process.

For instance, since the problem states nickels are ten less than twice the pennies, we have:

• \(n = 2p - 10\)

Similarly, dimes being twenty less than three times the pennies gives us:

• \(d = 3p - 20\)

By turning word problems into algebraic expressions, we can use these variable relationships to develop and solve correct equations.

Problem-Solving in Algebra

Problem-solving in algebra involves a series of logical steps to reach a solution. It requires careful analysis of the given problem and the relationships between various quantities. Here's how we tackled Maria's coin problem effectively:

• Understand the Problem: Begin by comprehending what is being asked. Identify the knowns and the unknowns.
• Define Variables: Assign variables to unknown quantities to create equations. Maria's problem required us to set \(p\), \(n\), and \(d\) for pennies, nickels, and dimes.
• Create Equations: Use the relationships described in the problem to form equations. For example, knowing total coins gives us \(p + n + d = 150\).
• Substitute and Simplify: Substitute expressions found from relationships, such as \(n = 2p -10\) and \(d = 3p - 20\), back into your main equation to simplify.
• Solve and Verify: Solve these equations to find the values of variables and verify by substituting them back into the original context to ensure they satisfy the conditions.

This process showcases a structured approach to tackling algebraic word problems, making it less daunting for students by providing a step-by-step methodology for problem-solving.