Problem 22 - Prealgebra

Question

Solve each equation. Check your solution. $$n-\frac{3}{8}=\frac{1}{6}$$

Step-by-Step Solution

Answer

The solution is \(n = \frac{13}{24}\).

Step 1: Isolate n

To solve for \(n\), first isolate it on one side of the equation. We have the equation \(n - \frac{3}{8} = \frac{1}{6}\). Add \(\frac{3}{8}\) to both sides to isolate \(n\): \[ n = \frac{1}{6} + \frac{3}{8} \]

Step 2: Find a Common Denominator

The fractions on the right side of the equation \(\frac{1}{6} + \frac{3}{8}\) have different denominators. We need a common denominator to add them. The least common denominator of 6 and 8 is 24. Convert each fraction: \[ \frac{1}{6} = \frac{4}{24} \quad \text{and} \quad \frac{3}{8} = \frac{9}{24} \]

Step 3: Add the Fractions

Now that the fractions have a common denominator, add them together: \[ \frac{4}{24} + \frac{9}{24} = \frac{13}{24} \] So, \(n = \frac{13}{24}\).

Step 4: Check the Solution

Substitute \(n = \frac{13}{24}\) back into the original equation to verify: \[ \frac{13}{24} - \frac{3}{8} = \frac{13}{24} - \frac{9}{24} = \frac{4}{24} = \frac{1}{6} \] The left side equals the right side of the equation, confirming that \(n = \frac{13}{24}\) is correct.

Key Concepts

Fractions

Fractions are a way to represent parts of a whole. They consist of a numerator, which is the top number, and a denominator, which is the bottom number. For example, in the fraction \( \frac{3}{8} \), *3* is the numerator, and *8* is the denominator.
Understanding fractions is crucial because they allow us to accurately represent amounts that are less than a whole number. Fractions are especially important in many areas of math, such as the problem we are working with here, \( n - \frac{3}{8} = \frac{1}{6} \).

• **Numerator**: Part of the fraction above the line—it represents how many parts we have.
• **Denominator**: Part of the fraction below the line—it signifies the total number of equal parts.

Recognizing how to manipulate and work with fractions is essential. Operations like addition, subtraction, multiplication, and division require a clear understanding of how fractions work. To perform these operations, attention must be paid to how the fractions relate to one another, particularly when dealing with equations.

Common Denominator

Finding a common denominator is a key step when adding or subtracting fractions. It involves finding a shared multiple of the denominators of the fractions you are working with. This ensures that the fractions have the same base, enabling us to perform operations with them directly.
In our exercise, we have two fractions: \( \frac{1}{6} \) and \( \frac{3}{8} \). The denominators are 6 and 8. The least common denominator here is 24, which is the smallest number into which both 6 and 8 can divide evenly.
Here’s how to convert each fraction to have this common denominator:

• Convert \( \frac{1}{6} \) to \( \frac{4}{24} \) because \( 6 \times 4 = 24 \) and the numerator follows suit: \( 1 \times 4 = 4 \).
• Convert \( \frac{3}{8} \) to \( \frac{9}{24} \) because \( 8 \times 3 = 24 \) and the numerator: \( 3 \times 3 = 9 \).

Once fractions share the same denominator, they can easily be added: \( \frac{4}{24} + \frac{9}{24} = \frac{13}{24} \). Converting to a common denominator is often a required step in solving fractional equations.

Checking Solutions

Checking your solution is a critical final step in solving an equation, as it confirms the accuracy of your answer.
When you solve for \(n\) and find \(n = \frac{13}{24}\), it's important to substitute back into the original equation to ensure that both sides of the equation balance. In mathematical terms, we check whether\[ \frac{13}{24} - \frac{3}{8} = \frac{1}{6} \].
To do this effectively, follow these steps:

• Replace \(n\) with your solution: \( \frac{13}{24} \).
• Convert \(\frac{3}{8}\) into the same denominator, 24, making it \(\frac{9}{24}\).
• Subtract: \( \frac{13}{24} - \frac{9}{24} = \frac{4}{24} \).

Since \( \frac{4}{24} \) simplifies to \( \frac{1}{6} \), the left side equals the right side of the original equation, so your solution for \( n \) is confirmed. Checking solutions reinforces your understanding and catches errors, ensuring that you can always be confident in your mathematical proficiency.