Problem 27 - Elementary Algebra

Question

Solve each of the equations. $$\frac{n}{150-n}=\frac{1}{2}$$

Step-by-Step Solution

Answer

The solution is \( n = 50 \).

Step 1: Cross-Multiply

The given equation is \( \frac{n}{150-n} = \frac{1}{2} \). To eliminate the fractions, cross-multiply: \( n \times 2 = (150-n) \times 1 \). This results in the equation \( 2n = 150 - n \).

Step 2: Isolate the Variable

Add \( n \) to both sides of the equation to combine like terms: \( 2n + n = 150 \). This simplifies to \( 3n = 150 \).

Step 3: Solve for \( n \)

Divide both sides of the equation by 3 to solve for \( n \): \( n = \frac{150}{3} \). This simplifies to \( n = 50 \).

Step 4: Verify the Solution

Substitute \( n = 50 \) back into the original equation to ensure it satisfies the condition: \( \frac{50}{150-50} = \frac{50}{100} = \frac{1}{2} \). The original equation is satisfied, confirming the solution is correct.

Key Concepts

Cross-Multiplication

When solving fractions in an equation, cross-multiplication is a handy tool that can help simplify the process. Imagine you have an equation like \( \frac{n}{150-n} = \frac{1}{2} \). The idea here is to eliminate the fractions by multiplying each denominator by the numerator on the opposite side of the equation.
This gives rise to a new equation without fractions: \( n \times 2 = (150-n) \times 1 \).
Cross-multiplication transforms a fraction-based problem into a simple algebraic equation, making it easier to solve.

• Multiply the numerator of one fraction by the denominator of the other fraction.
• Do the same for the other side.

This will leave you with a regular algebraic equation that's much simpler to handle. It's an essential step when dealing with equations containing fractions.

Isolate the Variable

To solve any equation, the main goal is often to isolate the variable—get it by itself on one side of the equation. After cross-multiplying, you end up with an equation like \( 2n = 150 - n \).
At this stage, you want to move all the terms involving your variable, \( n \), to one side.
The first thing you can do here is add \( n \) to both sides of the equation. Doing so gives \( 2n + n = 150 \) which simplifies to \( 3n = 150 \).

• Combine like terms, often by adding or subtracting terms from both sides.
• Get the variable by itself by using inverse operations, like addition, subtraction, multiplication, or division.

In this way, you slowly but surely isolate \( n \) by performing simple operations to simplify the equation further.

Verify the Solution

Once you've isolated the variable and found a solution, your work isn't quite done yet. It’s time to verify the solution to ensure it is correct. For our equation \( n = 50 \), we substitute back into the original equation to check our work.
Plug \( n = 50 \) into \( \frac{n}{150-n} = \frac{1}{2} \), which gives us \( \frac{50}{150-50} = \frac{50}{100} = \frac{1}{2} \).
Notice that both sides are equal, meaning the solution satisfies the original equation.

• Substitute your solution back into the original equation.
• Check to see if both sides of the equation are equal.

Verifying your solution ensures that you haven’t made a miscalculation and that the solution is indeed accurate. This step confirms your answer and keeps your work reliable.