Problem 29 - Elementary Algebra

Question

Solve each of the equations. $$\frac{300-n}{n}=\frac{3}{2}$$

Step-by-Step Solution

Answer

The solution is \(n = 120\).

Step 1: Understand the Problem

The problem asks us to solve the equation \(\frac{300-n}{n}=\frac{3}{2}\). This means we need to find the value of the variable \(n\) that makes this equation true.

Step 2: Cross-Multiply to Clear Fractions

To solve \(\frac{300-n}{n}=\frac{3}{2}\), we first multiply both sides by \(n\) and \(2\) to eliminate fractions: \(2(300-n)=3n\).

Step 3: Distribute and Simplify

Distribute \(2\) on the left side: \(2 \times 300 - 2 \times n = 3n\). This simplifies to \(600 - 2n = 3n\).

Step 4: Move Terms to Isolate Variable

To solve for \(n\), get all the \(n\) terms on one side of the equation and constants on the other: \(600 = 3n + 2n\), which simplifies to \(600 = 5n\).

Step 5: Solve for \(n\)

Divide both sides by \(5\) to isolate \(n\): \(n = \frac{600}{5}\). This simplifies to \(n = 120\).

Key Concepts

Understanding Cross-Multiplication

When you come across an equation with two fractions on opposite sides, like \( \frac{300-n}{n} = \frac{3}{2} \), cross-multiplication is a powerful technique to solve it. **Cross-multiplication** allows you to eliminate those fractions altogether. The process begins by multiplying the numerator of one fraction by the denominator of the other fraction and vice versa. Here’s how it works:

• Take \( 2 \), the denominator of the right fraction, and multiply it by \( 300-n \), the entire numerator of the left fraction, resulting in \( 2(300-n) \).
• Then take \( n \), the denominator of the left fraction, and multiply it by \( 3 \), the numerator on the right, resulting in \( 3n \).

This results in the equation \( 2(300-n)=3n \). Cross-multiplication has removed the fractions, making it easier to work with the equation.

How to Isolate Variables

Once the equation has been simplified, the next step is to **isolate the variable**. In our equation \( 2(300-n)=3n \), we need to solve for \( n \). This process involves moving all the terms containing the variable onto one side of the equation. Here’s how you can do it:

• First, distribute the \( 2 \) on the left side: \( 600 - 2n = 3n \).
• Next, you want to move all terms with \( n \) to one side. Add \( 2n \) to both sides to get \( 600 = 3n + 2n \).
• This results in the simpler equation \( 600 = 5n \).

The variable \( n \) is nearly by itself. This is our goal when isolating variables, to have \( n \) appear alone on one side of the equation. Once achieved, you'll be in a position to finalize the calculation and solve for \( n \).

Simplifying Algebraic Expressions

The final step often involves **algebraic simplification**. After isolating the variable in \( 600 = 5n \), you can perform a simple division to completely solve for \( n \). Here’s what to do next:

• Divide both sides of the equation by \( 5 \), the coefficient of \( n \): \( n = \frac{600}{5} \).
• This results in \( n = 120 \), which is your solution!

Simplifying algebraic expressions involves performing logical operations like addition, subtraction, multiplication, or division to make the equation more straightforward. It’s all about making the expression as simple as possible, so you find the answer with ease. Now, you have successfully solved for \( n \) using cross-multiplication, variable isolation, and algebraic simplification. This approach is versatile and very handy for solving various algebraic equations.