Problem 41 - Elementary Algebra

Question

Set up a proportion and then solve. If the ratio of female to male students at the college is 6 to 5 , then determine the number of male students out of 11,000 total students.

Step-by-Step Solution

Answer

There are 5,000 male students.

Step 1: Setting Up the Ratio

Firstly, we know the ratio of female to male students is 6 to 5, which can be written as a fraction \( \frac{6}{5} \). This means for every 11 students (6 females + 5 males), 5 are male.

Step 2: Express the Total as Proportions

The total number of students is given as 11,000. We express the number of female and male students using the given ratio. If \( x \) represents the number of males, then \( \frac{6}{5} = \frac{11,000 - x}{x} \).

Step 3: Formulating an Equation

Using the cross-multiplication method on the proportion, we get the equation: \( 6x = 5(11,000 - x) \).

Step 4: Solving for Number of Male Students

Expand the equation to get \( 6x = 55,000 - 5x \). Then, combine like terms to obtain \( 11x = 55,000 \). Solving for \( x \), we divide both sides by 11, resulting in \( x = 5,000 \).

Step 5: Conclusion

The number of male students out of 11,000 total students is 5,000.

Key Concepts

Understanding Ratios

Ratios are a way to express the relationship between two numbers or quantities. In the context of our problem, we're examining a specific ratio of female to male students, which is given as 6 to 5. This ratio indicates that for every 11 parts (6 parts female and 5 parts male), 5 parts are male. To better understand, think of it as a simplification of the real-world scenario, making it easier to calculate without knowing every detail. It's like saying "for every full pie of students, there are 5 slices of male students." Furthermore, ratios can be written in different forms:

• As a fraction: \( \frac{6}{5} \)
• Using a colon: 6:5
• In words: 6 to 5

Whichever format is used, the essence remains – showing how two or more quantities compare.

Introduction to Cross-Multiplication

Cross-multiplication is a mathematical technique used to solve equations that involve two fractions set equal to each other. In our proportion, it's used to solve the ratio of students effectively. When you set up a proportion like \( \frac{6}{5} = \frac{11,000 - x}{x} \), you effectively create two fractions that denote the same value. Here's where cross-multiplication comes in handy. This method involves:

• Multiplying the numerator of the first fraction (6) by the denominator of the second fraction (\( x \))
• Multiplying the numerator of the second fraction (\( 11,000 - x \)) by the denominator of the first fraction (5)

This leads to the equation: \( 6x = 5(11,000 - x) \). Such a method helps simplify the solution process, reducing potential errors in algebraic manipulations. By cross-multiplying, you transform the proportion into a standard algebraic equation, which is often easier to solve.

Solving Algebraic Equations

An algebraic equation is a mathematical statement composed of algebraic expressions, often containing one or more variables that you need to solve. In many exercises, as in this one, they form the critical groundwork for determining unknown values.In our equation, we derived it from the proportion: \( 6x = 5(11,000 - x) \). To solve it, we expand and simplify the equation through standard algebraic methods:

• Distribute \( 5 \) into \( 11,000 - x \), transforming it into \( 6x = 55,000 - 5x \)
• Add\( 5x \) to both sides to organize like terms, resulting in \( 11x = 55,000 \)
• Finally, divide both sides by \( 11 \) to solve for \( x \): \( x = 5,000 \)

These steps show how an algebraic approach leads straight to the answer, demonstrating the beauty and efficiency of mathematical processes in offering clarity and correctness in problem-solving.