Problem 73 - Elementary Algebra

Question

Set up an algebraic equation and then solve. Cathy has to deposit \(\$ 410\) worth of five- and ten-dollar bills. She has 1 fewer than three times as many tens as she does five-dollar bills. How many of each bill does she have to deposit?

Step-by-Step Solution

Answer

Cathy has 12 five-dollar bills and 35 ten-dollar bills.

Step 1: Define Variables

Let \( x \) represent the number of five-dollar bills that Cathy has. The problem states that Cathy has 1 fewer than three times as many ten-dollar bills as five-dollar bills. Therefore, we can express the number of ten-dollar bills as \( 3x - 1 \).

Step 2: Set Up the Equation

Cathy's total deposit of \$410 consists of her five- and ten-dollar bills. We can express this relationship as an equation: \( 5x + 10(3x - 1) = 410 \).

Step 3: Simplify the Equation

Expand and simplify the equation: \( 5x + 30x - 10 = 410 \). This simplifies further to \( 35x - 10 = 410 \).

Step 4: Solve for \( x \)

Add 10 to both sides of the equation to isolate the term with \( x \): \( 35x = 420 \). Then divide both sides by 35 to find \( x \): \( x = \frac{420}{35} = 12 \). Thus, Cathy has 12 five-dollar bills.

Step 5: Calculate Number of Ten-Dollar Bills

Use the value of \( x \) to find the number of ten-dollar bills: \( 3x - 1 = 3(12) - 1 = 36 - 1 = 35 \). Thus, Cathy has 35 ten-dollar bills.

Key Concepts

Problem Solving

Approaching a problem with a clear strategy can make solving algebraic equations much easier. In the given problem, we want to find out how many five-dollar and ten-dollar bills Cathy has.
Understanding the problem is the first step:

• Identify what is given and what needs to be found.
• List down known quantities and relationships between them.
• Read the problem statement carefully to ensure no detail is overlooked.

This particular problem tells us the total value of the bills and gives a relation between the numbers of each type of bill.
By breaking down the problem in manageable parts, we can create precise steps to reach a solution.

Variables

In algebra, variables are symbols that represent unknown values. They are crucial for forming equations and ultimately solving them. In this problem, let's make Cathy’s money situation clearer.
We define:

• Let \( x \) be the number of five-dollar bills.
• The number of ten-dollar bills can then be expressed in terms of \( x \), as \( 3x - 1 \).

Choosing the right variable and setting up expressions properly allows us to reorganize the problem into algebraic equations.
With this foundation, we can use arithmetic operations to find unknown values.

Simplifying Equations

Simplifying equations involves several arithmetic operations to make them easier to solve. By following these steps, we break down complex equations into simpler ones. First, construct the equation from the problem's narrative:
\( 5x + 10(3x - 1) = 410 \).
Expanding and combining common terms gives us:

• Multiply \( 10 \) by \( 3x - 1 \) to get \( 30x - 10 \).
• Add \( 5x \) to \( 30x - 10 \) to have \( 35x - 10 = 410 \).

From there, isolate the variable term, making it easier to solve the equation in subsequent steps.

Linear Equations

Linear equations are equations of the first degree, which means the highest power of the variable is one. This problem showcases how to work with linear equations, specifically in the form \( ax + b = c \).
Let’s break it down by solving for \( x \):

• Add \( 10 \) to both sides of the equation: \( 35x = 420 \).
• Divide both sides by \( 35 \) to isolate \( x \): \( x = 12 \).

This yields the solution for one variable. From the solution, you can determine related values like the number of ten-dollar bills.
Linear equations are fundamental and often seen in various real-world scenarios, making mastering them highly beneficial.