Problem 39 - Elementary Algebra

Question

Set up a proportion and then solve. Out of 460 students surveyed, the ratio to support the student union remodel project was 3 to 5 . How many students were in favor of the remodel?

Step-by-Step Solution

Answer

276 students were in favor of the remodel.

Step 1: Understand the Given Ratio

The problem states that for every 3 students who support the remodel, there are 5 students overall (including both those who support and those who do not support the remodel). This gives us a ratio of 3:5.

Step 2: Set Up the Proportion

A proportion relates two ratios or fractions. Based on the ratio given, for every 5 students, 3 support the remodel. Thus, we can set up the proportion: \( \frac{3}{5} = \frac{x}{460} \), where \( x \) is the number of students in favor of the remodel out of the total 460 surveyed.

Step 3: Cross-Multiply to Solve the Proportion

Cross-multiply to solve for \( x \):\[3 \times 460 = 5 \times x\] This simplifies to \( 1380 = 5x \).

Step 4: Solve for x

To find \( x \), divide both sides of the equation by 5:\[x = \frac{1380}{5} = 276\]

Step 5: Verify the Solution

Double-check the solution by plugging \( x = 276 \) back into the context of the problem. The ratio \( \frac{276}{460} \) simplifies to \( \frac{3}{5} \), confirming that 276 students are in favor of the remodel, consistent with the given ratio.

Key Concepts

Understanding Ratios

Ratios are a way of comparing two quantities. In the context of this exercise, the ratio is given as 3:5. This can be interpreted as, for every 3 students who support the remodel, there is a group of 5 students total. It’s a simplified way to describe a part-to-whole relationship. The first number (3) refers to the part, and the second number (5) refers to the whole, including both supporters and non-supporters.

Ratios can be expressed in different forms, such as a fraction, "3 to 5," or 3:5. These are all equivalent representations. Understanding this helps in setting up and solving problems because it allows you to work with proportions, as seen in this exercise. By thinking of the ratio as a fraction, it becomes easier to set up equations that model real-world situations.

Cross-Multiplication Technique

Cross-multiplication is a helpful method used in solving proportions. A proportion is an equation that states two ratios are equivalent. In our exercise, the proportion is set as \(\frac{3}{5} = \frac{x}{460}\). To find the unknown value \(x\), we use cross-multiplication.

The process involves multiplying the numerator of one ratio by the denominator of the other, then setting the products equal. For this equation, multiply 3 by 460 and set it equal to 5 times \(x\), producing the equation \(3 \times 460 = 5 \times x\). This simplifies the proportion for easier solving. This step is crucial in converting a proportion to an equation which can be solved using basic arithmetic operations.

This technique effectively turns the problem into a simpler equation, allowing for easier calculation and interpretation of the results.

Solving Equations

Once you've set up an equation through cross-multiplication, solving it becomes a matter of basic algebra. In this exercise, the equation derived from cross-multiplication is \(1380 = 5x\). The goal is to isolate \(x\) on one side of the equation to find its value.

To do this, divide both sides of the equation by 5: \(x = \frac{1380}{5}\). This calculation results in \(x = 276\), meaning 276 students favor the remodel. Solving equations often involves operations such as addition, subtraction, multiplication, and division. By systematically applying these operations, you can isolate the unknown variable and find its value.

It's important to verify your solution by checking if it makes sense in the context of the original problem. Plugging \(x = 276\) back into the given ratio confirms that the solution is consistent with the stated proportion, ensuring it's a valid solution.