Problem 34 - Intermediate Algebra

Question

In a recent survey, one thousand registered voters were asked about their political preferences. The number of males in the survey was five less than one-half of the number of females. Find the number of males in the survey.

Step-by-Step Solution

Answer

There are 330 males in the survey.

Step 1: Define Variables

Let \( F \) represent the number of females in the survey. Since the total number of voters is 1000, the equation becomes \( F + M = 1000 \), where \( M \) is the number of males.

Step 2: Express Males in Terms of Females

According to the problem, the number of males \( M \) is five less than one-half of the number of females. This can be expressed as \( M = \frac{1}{2}F - 5 \).

Step 3: Substitute and Solve

Substitute the expression for \( M \) from the previous step into the equation \( F + M = 1000 \), giving \( F + \left( \frac{1}{2}F - 5 \right) = 1000 \). Simplify and solve for \( F \):\[ F + \frac{1}{2}F - 5 = 1000 \] \[ \frac{3}{2}F - 5 = 1000 \] \[ \frac{3}{2}F = 1005 \] \[ F = \frac{1005 \times 2}{3} \] \[ F = 670 \].

Step 4: Calculate the Number of Males

Now that we know \( F = 670 \), use the equation \( M = \frac{1}{2}F - 5 \) to find \( M \):\[ M = \frac{1}{2} \times 670 - 5 \] \[ M = 335 - 5 \] \[ M = 330 \].

Key Concepts

Understanding Systems of Equations

A system of equations is essentially a set of two or more equations with the same variables. In this exercise, we are dealing with two variables: the number of males and the number of females who took part in the survey.

We create two different equations based on the information provided:

• The total number of voters is set at 1000: \( F + M = 1000 \).
• The number of males is described as five less than one-half the number of females: \( M = \frac{1}{2}F - 5 \).

Each equation provides a different piece of information about the same scenario. Collectively, these equations form what is known as a system of equations.

With these two equations, we can find a specific solution for the values of \( F \) and \( M \). By solving the system, we understand how both quantities relate to each other in this particular situation.

Exploring Variable Substitution

Variable substitution is a common technique used to solve systems of equations. It involves replacing one variable with an expression involving another variable from another equation. In this exercise, we express \( M \) in terms of \( F \) from the second equation: \( M = \frac{1}{2}F - 5 \).

We then substitute this expression into the first equation: \( F + M = 1000 \). During this substitution, \( M \) is replaced with \( \frac{1}{2}F - 5 \), resulting in:

• \( F + \left( \frac{1}{2}F - 5 \) = 1000

This transformation reduces the system to a single equation in terms of one variable (\( F \)), making it simpler to solve.

The key advantage of variable substitution is that it simplifies solving the system by reducing the number of variables and equations you need to juggle at once.

Simplifying with Basic Arithmetic Operations

Basic arithmetic operations such as addition, subtraction, multiplication, and division play a fundamental role in solving algebra word problems. After substituting the equation for \( M \) into \( F + M = 1000 \), the challenge is to simplify and solve for one variable.

First, combine like terms in the equation \( F + \frac{1}{2}F - 5 = 1000 \) to simplify it:

• \( \frac{3}{2}F - 5 = 1000 \).

Next, eliminate the subtraction by adding 5 to both sides:

• \( \frac{3}{2}F = 1005 \).

Then, proceed with multiplication or division to isolate \( F \):

• Multiply both sides by \( \frac{2}{3} \) to solve for \( F \): \( F = \frac{1005 \times 2}{3} \).

Finally, we find \( F = 670 \).

These steps illustrate how basic arithmetic operations are used effectively to calculate and deduce the number of males and females in the survey.