Problem 34 - Prealgebra

Question

Solve each equation. Check your solution. $$\frac{1}{3} n=\frac{2}{9}$$

Step-by-Step Solution

Answer

The solution to the equation is \( n = \frac{2}{3} \).

Step 1: Understand the Equation

We are given the equation \( \frac{1}{3}n = \frac{2}{9} \). This is a simple linear equation involving a fraction.

Step 2: Clear the Fraction by Multiplying

To remove the fraction, multiply both sides of the equation by 3, the denominator of the fraction multiplying \( n \). This gives us \( n = \frac{2}{9} \times 3 \).

Step 3: Simplify the Right Side

Simplify the expression on the right side: \( \frac{2}{9} \times 3 = \frac{6}{9} \). Since \( \frac{6}{9} \) can be simplified, reduce it to \( \frac{2}{3} \). Thus, \( n = \frac{2}{3} \).

Step 4: Verify Your Solution

Substitute \( n = \frac{2}{3} \) back into the original equation: \( \frac{1}{3} \times \frac{2}{3} \). Simplify this to check: \( \frac{2}{9} \). Since both sides of the original equation are equal, \( n = \frac{2}{3} \) is correct.

Key Concepts

Fractions

Fractions are a way of expressing numbers that are not whole. They consist of a numerator, the top part of the fraction, and a denominator, the bottom part. Understanding fractions is crucial since they often appear in various mathematical problems, including equations.
In our equation, we deal with a fraction: \(\frac{1}{3}n = \frac{2}{9}\). The fraction \(\frac{1}{3}\) is multiplied by \(n\), while another fraction, \(\frac{2}{9}\), stands alone on the other side of the equation. To solve problems involving fractions, one key technique is making them easier to manage. Sometimes, this involves converting them to a format that is easier to work with, like whole numbers.
When fractions share a similar denominator or can be simplified, solving the equation becomes straightforward. This is why removing fractions by multiplying or simplifying them is a beneficial strategy. Developing comfort and skill with fractions opens doors to solving more intricate mathematical problems.

Solving Equations

Solving equations refers to finding the value of the variable that makes the equation true. Equations can seem complex, especially when they involve fractions or multiple steps.
The equation in our exercise is originally presented as \(\frac{1}{3}n = \frac{2}{9}\). To solve it, we strategically eliminate the fraction where possible. A common technique is multiplying both sides of the equation by the denominator of the fraction on the side where the variable exists. Here, we multiply by 3 to clear the fraction:

• This transforms the left side of the equation into \(n\).
• Then multiply the other side: \(\frac{2}{9} \times 3 = \frac{6}{9}\).

Now, solving the equation becomes easier as we've transformed it into one with simpler numbers or fractions, facilitating further steps to finding the solution.

Simplifying Expressions

Simplifying expressions is a key mathematical skill that involves reducing expressions to their most basic form, removing any unnecessary complexity. For example, \(\frac{6}{9}\) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3.
In this exercise, after multiplying to eliminate fractions, we were left with the expression \(\frac{6}{9}\). This was simplified to \(\frac{2}{3}\) by recognizing that both 6 and 9 are divisible by 3:

• Divide the numerator by 3: \(6 \div 3 = 2\).
• Divide the denominator by 3: \(9 \div 3 = 3\).

This simplification process is crucial as it not only makes expressions more understandable but also checks for accuracy when solving or verifying equations. Simplifying helps ensure that final answers are accurate and precise.