Problem 68 - Elementary Algebra

Question

A marketing company surveys 1,200 people. They surveyed twice as many females as males. How many males and females did they survey?

Step-by-Step Solution

Answer

400 males and 800 females.

Step 1: Define the Variables

Let the number of males surveyed be represented by the variable \(m\). According to the problem, the number of females surveyed is twice the number of males, so we represent the number of females as \(2m\).

Step 2: Set Up the Equation

We know the total number of people surveyed is 1,200. Therefore, we can set up the equation: \[m + 2m = 1200\]

Step 3: Combine Like Terms

Combine the terms on the left side of the equation: \[3m = 1200\]

Step 4: Solve for \(m\)

Divide both sides of the equation by 3 to solve for \(m\): \[m = \frac{1200}{3} = 400\]

Step 5: Find the Number of Females

Since the number of females is twice the number of males, substitute \(m = 400\) into the expression for females: \[2m = 2 \times 400 = 800\]

Key Concepts

Defining Variables

To solve algebra word problems effectively, it is crucial to first define the variables clearly. In our exercise, we need to calculate the number of males and females surveyed by a marketing company. Let’s represent these unknowns with variables. Suppose we let the number of males be denoted by the variable \( m \). According to the problem, the number of females surveyed is twice the number of males. Therefore, we can represent the number of females as \( 2m \). Defining variables helps us translate the words into mathematical expressions, making the problem easier to solve.

Setting Up Equations

After defining the variables, the next step is to set up an equation based on the given information. From the problem, we know the total number of surveyed individuals is 1,200. We also identified that the number of males is \( m \) and the number of females is \( 2m \). Since the total number of people surveyed is the sum of males and females, we can write this relationship as an equation: \[ m + 2m = 1200 \]. Setting up equations from word problems is key in algebra as it converts verbal information into a solvable mathematical format.

Combining Like Terms

Once we set up the equation, we need to simplify it by combining like terms. In our equation \( m + 2m = 1200 \), we add the terms with the same variable. The terms \( m \) and \( 2m \) are like terms as they both contain the variable \( m \). Combining them gives us: \[ 3m = 1200 \]. This simplification step makes the equation easier to solve by reducing the number of terms on one side.

Solving Linear Equations

With the simplified equation \[ 3m = 1200 \], we can now solve for the variable \( m \). To isolate \( m \), we divide both sides of the equation by 3: \[ m = \frac{1200}{3} = 400 \]. Therefore, the number of males surveyed is 400. To find the number of females, we use our earlier expression for females: \[ 2m = 2 \times 400 = 800 \]. Solving linear equations involves reversing the operation performed on the variable to isolate it. This step is essential in determining the actual values represented by the variables.