Problem 59 - Algebra for College Students

Question

Set up an algebraic equation and solve each problem. Suppose that in a certain precinct, 1150 people voted in the last presidential election. If the ratio of female voters to male voters was 3 to 2 , how many females and how many males voted?

Step-by-Step Solution

Answer

There were 690 female voters and 460 male voters.

Step 1: Understand the Ratio

The problem states that the ratio of female voters to male voters is 3 to 2. This means that for every 3 female voters, there are 2 male voters.

Step 2: Define the Variables

Let the number of female voters be represented by \( F \) and the number of male voters by \( M \). According to the ratio, \( F:M = 3:2 \). This implies \( \frac{F}{M} = \frac{3}{2} \).

Step 3: Express One Variable in Terms of the Other

From the ratio \( \frac{F}{M} = \frac{3}{2} \), we can express \( F \) in terms of \( M \) as \( F = \frac{3}{2}M \).

Step 4: Set Up the Equation

We know the total number of voters is 1150: thus, \( F + M = 1150 \). Substitute \( F = \frac{3}{2}M \) into this equation: \( \frac{3}{2}M + M = 1150 \).

Step 5: Solve for \( M \)

Combine terms: \( \frac{3}{2}M + \frac{2}{2}M = 1150 \). That simplifies to \( \frac{5}{2}M = 1150 \). Solve for \( M \) by multiplying both sides by \( \frac{2}{5} \): \( M = 1150 \times \frac{2}{5} = 460 \).

Step 6: Solve for \( F \)

Substitute \( M = 460 \) into \( F = \frac{3}{2}M \): \( F = \frac{3}{2} \times 460 = 690 \).

Step 7: Verify the Solution

The total number of voters is \( 460 + 690 = 1150 \), which matches the given total. The ratio \( \frac{690}{460} = \frac{3}{2} \) is consistent with the problem statement. Thus, our solution is correct.

Key Concepts

Ratios in Algebra

In algebra, ratios are used to compare two or more quantities. Ratios can be a powerful way to express relationships between numbers, and they allow us to solve problems involving relative quantities. A ratio in algebra can be translated into an equation which reflects how these quantities change together. In the exercise we dealt with, the ratio was given as 3 to 2. This means that for every 3 female voters, there are 2 male voters. When adjusting ratios into algebraic terms:

• You interpret the given ratio as a fraction, like \( \frac{3}{2} \).
• This means the first value is numerically equivalent to the fraction times the second value.

Understanding how to manipulate ratios is key in solving real-world algebra problems as it simplifies complex comparisons into actionable equations.

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding unknown values that make a mathematical statement true. This exercise required finding how many female and male voters there were, given their total and their ratio. The process is essentially about setting up an equation that represents the relationship and solving for the unknowns. Here, we used these steps:

• First, express one variable in terms of the other using the ratio: \( F = \frac{3}{2}M \).
• Next, set up an equation with the known total: \( F + M = 1150 \).
• Substitute the expression for one of the variables: \( \frac{3}{2}M + M = 1150 \).
• Finally, solve the resulting equation for one variable, then back-solve for the other.

Taking these steps one by one helps decomposing a complex problem into manageable parts. Ultimately, successful equation solving depends on understanding the problem, translating it into mathematical terms, and then manipulating the algebraic statements to find answers.

Variables in Equations

Variables are symbols used to represent unknowns in equations. They are fundamental to writing and solving algebraic equations. In our exercise, the variables \( F \) and \( M \) represented the number of female and male voters, respectively.

• Variables allow us to express complex relationships simply and flexibly.
• We used the variable \( F \) for females and \( M \) for males.
• This abstraction made it easier to set up the equation \( F + M = 1150 \) based on the given total voters.

The power of variables lies in their ability to transform verbal problems into solvable mathematical equations. By substituting variables for unknown quantities, and using relationships like ratios, we gain a clear path to finding specific values. Mastering the use of variables helps in tackling various algebraic challenges efficiently.