Problem 8 - Prealgebra

Question

Find each sum or difference. Write in simplest form. \(-\frac{1}{8}-\frac{5}{8}\)

Step-by-Step Solution

Answer

The simplified result of \(-\frac{1}{8}-\frac{5}{8}\) is \(-\frac{3}{4}\).

Step 1: Identify the Operation

The problem requires us to subtract two fractions: \(-\frac{1}{8}\) and \(\frac{5}{8}\). Both fractions have the same denominator, so we can perform the operation directly on the numerators.

Step 2: Subtract the Numerators

Since the denominators are the same (8), subtract the numerator of the second fraction from the numerator of the first fraction: \(-1 - 5 = -6\).

Step 3: Simplify the Fraction

The result of the subtraction is \(-\frac{6}{8}\). We simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. Thus, \(-\frac{6}{8} = -\frac{3}{4}\).

Key Concepts

Simplifying Fractions

When you work with fractions, you'll often encounter the task of simplifying them. Simplifying involves reducing the fraction to its simplest form, where the numerator and the denominator have no common factors other than 1. For example, simplifying a fraction like \(-\frac{6}{8}\) involves finding the greatest common divisor (GCD) of 6 and 8, which is 2.
By dividing both the numerator and the denominator by their GCD, you decrease the values equally without changing the fraction's value. Hence, \(-\frac{6}{8}\) simplifies to \(-\frac{3}{4}\) because:

• Divide 6 and 8 by 2 (the GCD):
• 6 ÷ 2 = 3
• 8 ÷ 2 = 4

This reduced form is the simplest form of the fraction. Simplifying fractions is crucial as it makes the math easier to handle and understand.

Common Denominators

Common denominators come into play when you are performing operations such as addition or subtraction on fractions. The denominator is the bottom number of a fraction, indicating the total number of equal parts into which the whole is divided. In our original exercise: \(-\frac{1}{8} - \frac{5}{8}\), both fractions share the same denominator (8).
Having a common denominator is beneficial because it allows you to directly add or subtract the numerators while keeping the denominator constant. This simplifies the operation and ensures that you’re working with parts of the same size.
If two fractions do not have the same denominator, you must find a common denominator, usually the least common multiple (LCM) of the denominators. Once found, convert each fraction to an equivalent fraction with this common denominator before performing the operation. This ensures consistency and accuracy when calculating the result.

Negative Fractions

Negative fractions might seem tricky at first, but they're simple once you understand the concept of negative numbers in fractions. A negative fraction is a fraction with either the numerator or the denominator being negative, or sometimes both, to indicate a value less than zero. For example, \(-\frac{1}{8}\) means you're taking one part out of eight less than zero.
In subtraction, as shown in the exercise \(-\frac{1}{8} - \frac{5}{8}\), dealing with negative fractions involves applying the same rules as with normal numbers. You perform the operation on the numerators, remembering that subtracting a positive number from a negative results in a more negative number.

• Numerator calculation: \(-1 - 5 = -6\)
• Thus, giving you the fraction: \(-\frac{6}{8}\)

Handling negative fractions becomes straightforward once you familiarize yourself with the basic arithmetic of negative numbers.