Question

First solve the problem, and then enter your answer on the grid provided on the answer sheet. The instructions for entering your answers follow. On a particular college campus, there are two men for every three women. If the total number of men and women on campus is equal to 4,000 , how many more women are there on campus than men?

Short answer

There are 800 more women than men on campus.

Step-by-step solution

Define the Variables

Let's denote the number of men as \( m \) and the number of women as \( w \). According to the information given, the ratio of men to women is 2:3. This can be expressed as \( \frac{m}{w} = \frac{2}{3} \).

Express the Ratio as an Equation

The ratio \( \frac{m}{w} = \frac{2}{3} \) implies that \( 3m = 2w \). This equation helps us to relate the number of men to the number of women.

Use the Total Number Condition

The problem states the total number of men and women is 4,000. Therefore, we have the equation \( m + w = 4000 \).

Solve the System of Equations

We have two equations:1. \( 3m = 2w \)2. \( m + w = 4000 \)Substitute \( m = \frac{2}{3}w \) from the first equation into the second:\[\frac{2}{3}w + w = 4000\]Combine the terms:\[\frac{5}{3}w = 4000\]Solving for \( w \):\[w = 4000 \times \frac{3}{5} = 2400\]

Find the Number of Men

Using the number of women \( w = 2400 \), substitute back into the equation \( m = \frac{2}{3}w \):\[m = \frac{2}{3} \times 2400 = 1600\]

Calculate the Difference

The number of women minus the number of men is given by \( w - m = 2400 - 1600 = 800 \). This tells us there are 800 more women than men.

Key concepts

Understanding Ratio Problems

Ratio problems are a common type of mathematical challenge, and they're easier to tackle when you understand what a ratio is. A ratio is a way to compare two quantities by showing how many times one value contains or is contained within the other. In our college campus scenario, the ratio of men to women is given as 2:3. This means for every 2 men, there are 3 women. Ratios can help us transform a word problem into an algebraic format, simplifying the process of finding a solution.

To solve a ratio problem, you typically set up a relationship or equation between the two quantities. In this context, the relationship was expressed as \( \frac{m}{w} = \frac{2}{3} \), showing the ratio directly as a fraction. The further elaboration, converting this into the equation \( 3m = 2w \), helps use ratios to find exact numbers. Recognizing these relationships is crucial, especially for PSAT Math tasks where establishing the correct ratio can lead directly to a solution.

Solving with Algebraic Equations

Algebraic equations are mathematical statements that express the equality between two expressions. They're a foundational part of solving PSAT Math problems, especially when dealing with numbers indirectly given through conditions like ratios. From the step-by-step solution, you can see how algebra was applied.

We started with two main equations here: 1) the ratio expressed as \( 3m = 2w \) and 2) the total number of people on campus, formulated as \( m + w = 4000 \). These two equations together form a system of equations, which can be solved simultaneously.

By substituting part of one equation into another, we eliminate one variable, which simplifies the solution process. For example, substituting \( m = \frac{2}{3}w \) into the total equation gave a single equation in terms of \( w \). This step is crucial for simplifying complex problems down to easily manageable parts.

Tackling Word Problems in Math

Word problems are a staple of math exams, like the PSAT, because they test a wide range of skills — from reading comprehension to mathematical reasoning. In tackling word problems, it is essential to translate the words into mathematical language effectively.

Begin by identifying the variables — here, \( m \) for men and \( w \) for women — and understanding the relationships between them, such as ratios or sums. Next, form the equations based on these relationships. The problem on the college campus is a classic example, where reading "two men for every three women" and "total of 4,000" leads to the equations in the step-by-step solution.

Once equations are established, solving them with methods like substitution or elimination can efficiently lead to the solution. Remember, practicing word problems helps in recognizing patterns and common structures, making it much simpler to know where and how to begin.