Problem 8 - Prealgebra

Question

Find the following sums and differences, and reduce to lowest terms. (Add or subtract as indicated.) $$\frac{x}{8}-\frac{1}{8}$$

Step-by-Step Solution

Answer

The simplified form is \(\frac{x-1}{8}\).

Step 1: Identify Common Denominator

Both fractions in the expression \(\frac{x}{8}-\frac{1}{8}\) have the same denominator, which is 8. This allows us to directly combine the fractions by subtracting the numerators.

Step 2: Subtract the Numerators

Since the denominators are the same, subtract the numerators. The expression becomes \(\frac{x-1}{8}\).

Step 3: Simplify the Fraction

In this case, the fraction \(\frac{x-1}{8}\) is already in its simplest form, since the numerator and denominator have no common factors other than 1.

Key Concepts

Understanding Common Denominator

When working with fraction subtraction, one of the most crucial steps is identifying a common denominator. The denominator is the bottom part of a fraction and represents the total number of equal parts a whole is divided into. A common denominator between two fractions means the two fractions share the same total parts, which simplifies the process of combining (adding or subtracting) them.

In the exercise \(\frac{x}{8} - \frac{1}{8}\), notice that both fractions already have a denominator of 8. This saves the initial step of finding a similar base and allows us to move directly to the next step: subtracting the numerators. Having a common denominator is a fundamental prerequisite when performing operations with fractions, as it ensures they are compatible and can be easily combined.

Performing Numerator Subtraction

Once the common denominator is established, the focus shifts to the numerators—the top portions of the fractions. With fraction subtraction, the process involves subtracting the second numerator from the first, while keeping the common denominator constant.

In the exercise, we begin with the fractions \(\frac{x}{8}\) and \(\frac{1}{8}\). Given that both denominators are identical (8), we subtract the numerators: \(x - 1\). The resulting subtraction gives us the fraction \(\frac{x-1}{8}\). The step of numerator subtraction is crucial as it dictates the overall result of the subtraction, affecting the size and sign of the resulting fraction.

Simplifying Fractions Correctly

The final step in working with fractions like \(\frac{x-1}{8}\) is simplifying the fraction to its simplest form. Simplification involves reducing the fraction such that the numerator and denominator no longer have any common factors, other than 1.

In our exercise, \(\frac{x-1}{8}\), it is already evident that the expression is in its simplest form. This is because there are no apparent common factors between \(x-1\) and 8 other than 1. However, in different scenarios where both the numerator and the denominator share common factors, you would divide both by the greatest common factor to simplify the fraction.

Simplifying fractions is important as it presents the fraction in its most basic and easily understandable form, making further mathematical manipulations more straightforward.