Problem 245 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve. Brandon has a cup of quarters and dimes with a total value of \(\$ 3.80\). The number of quarters is four less than twice the number of dimes. How many quarters and how many dimes does Brandon have?

Step-by-Step Solution

Answer

Brandon has 12 quarters and 8 dimes.

Step 1 - Define Variables

Let the number of quarters be represented by \( q \) and the number of dimes be represented by \( d \).

Step 2 - Set Up Value Equation

Using the information given, the total value equation in terms of dimes and quarters can be written as: 0.25q + 0.10d = 3.80.

Step 3 - Set Up Quantity Relationship

According to the problem, the number of quarters is four less than twice the number of dimes, which can be written as \( q = 2d - 4 \).

Step 4 - Substitute Value

Substitute the expression for \( q \) from the relationship equation into the value equation: 0.25(2d - 4) + 0.10d = 3.80

Step 5 - Simplify Equation

Distribute and combine like terms: 0.50d - 1.00 + 0.10d = 3.80 which simplifies further to 0.60d - 1.00 = 3.80.

Step 6 - Solve for d

Add 1.00 to both sides of the equation to isolate the terms with \( d \): 0.60d = 4.80, then divide both sides by 0.60: \(d = 8\).

Step 7 - Solve for q

Use the value of \( d \) in the relationship equation to find \( q \): \( q = 2(8) - 4 \) which simplifies to \( q = 12\).

Step 8 - Verify Solution

Check the values in both equations. The total value of 12 quarters and 8 dimes is 12(0.25) + 8(0.10) which indeed equals 3.80, confirming the solution.

Key Concepts

system of equations

A system of equations is a set of two or more equations with the same variables. The goal is to find the values of these variables that satisfy all equations simultaneously. In this exercise, we are given two pieces of information:

• The total value of quarters and dimes.
• The relationship between the number of quarters and dimes.

To solve such problems, we need to translate these pieces of information into mathematical equations.

substitution method

The substitution method is a technique for solving a system of equations. Here is the process step by step:

• First, solve one of the equations for one variable in terms of the other variable.
• Substitute this expression into the other equation. This results in an equation with only one variable.
• Solve this single-variable equation to find the value of one variable.
• Use this value to find the value of the other variable from the first equation.

In our exercise, we solved the second equation for the number of quarters (q) and substituted this into the value equation.

coin word problems

Coin word problems often involve finding the number of coins of different denominations when given a total value and some relationship between the number of coins. Here's a simple strategy to follow:

• First, identify the types of coins and their values (quarters and dimes in this case).
• Then, write down what each coin type represents (let q be quarters and d be dimes).
• Translate the word problem into a system of equations, where one equation represents the total value and the other represents the quantity relationship.
• Solve the system of equations using a method like substitution or elimination.

solving linear equations

Solving linear equations is a fundamental skill in algebra. A linear equation is an equation in which the highest power of the variable is 1. When solving these equations, follow these steps:

• Simplify both sides of the equation by combining like terms and removing parentheses.
• Isolate the variable by performing the same operation on both sides of the equation (e.g., adding, subtracting, multiplying, or dividing).
• Check your solution by plugging it back into the original equation to ensure it satisfies the equation.

In our exercise, we isolated the variable d in the simplified equation and solved for it by performing addition and division.