Question

Universal Products has 78 employees. If twice as many women work for Universal as men, how many women work for Universal? A. 52 B. 42 C. 26 D. 16

Short answer

A. 52

Step-by-step solution

Write the equation for the total number of employees

The total number of employees is 78, which is the sum of women and men. We can represent this as an equation: \[M + W = 78\]

Write the equation for the ratio of women to men

According to the problem, there are twice as many women as men. We can represent this as an equation: \[W = 2M\]

Substitute the equation from step 2 into the main equation

Now, substitute the equation from step 2 into the main equation from step 1 to eliminate one variable: \[M + (2M) = 78\]

Solve for M

Combine the M terms: \[3M = 78\] Now, divide both sides of the equation by 3 to solve for M: \[M = \frac{78}{3}\] \[M = 26\]

Solve for W

Now, use the value of M to find the value of W by substituting M into the equation from step 2: \[W = 2(26)\] \[W = 52\] So, there are 52 women working for Universal Products. Therefore, the correct answer is: A. 52

Key concepts

Understanding Ratios

Ratios are a way to compare two quantities by showing how many times one value contains or is contained within the other. In our exercise, the problem states that there are twice as many women as there are men working at Universal Products. Expressing this condition as a ratio, we can say that for every man, there are two women.
Ratios can often help us find unknown quantities when combined with other known information. They are expressed in various forms such as "2:1" or "2 to 1" and can also be converted into equations for easier calculation. Understanding ratios is crucial because it allows you to represent relationships clearly and solve real-world problems with different quantities.

Forming and Using Equations

An equation is a mathematical statement that shows the equality of two expressions. It is a crucial tool in solving algebra word problems.
In our exercise, there are two main equations derived from the problem description:

• The total number of employees: \(M + W = 78\)
• The ratio of women to men: \(W = 2M\)

These equations allow us to find the unknown quantities by substitution or transformation techniques.
By substituting the second equation into the first one, we convert a two-variable problem into a single-variable problem, making it simpler to solve. Therefore, forming equations is a critical step that helps us translate real-world problems into solvable mathematical expressions.

Effective Problem Solving

Problem solving often involves breaking down a word problem into manageable steps. The goal is to find a logical path to arrive at the solution. In this exercise, the following method was used:

• Identify what you know and what you need to find out. Here, the known values are the total number of employees and the condition about the women's numbers.
• Convert the word problem into equations that express the given relationships.
• Simplify those equations to find the unknown variables step by step.
• Verify the results by checking if they fit the conditions given in the problem.

A structured approach to problem solving helps in organizing thoughts, ensuring that no details are missed, and leads to accurate solutions.

Basic Algebra in Action

Basic algebra involves using symbols and letters to represent numbers in mathematical expressions and equations. It's the foundation of higher-level mathematics and a valuable skill for solving everyday problems.
In this exercise, basic algebra allowed us to represent unknown numbers with letters like \(M\) for men and \(W\) for women. We used operations like addition, multiplication, and division to solve the equations.
The steps involved using substitution and simplification, which are essential techniques in algebra. These techniques allow you to reduce complex problems into simpler forms that are easier to solve. In essence, basic algebra turns word problems into mathematical puzzles, providing a straightforward pathway to uncover the answers.