Problem 7 - Elementary Algebra

Question

Reduce each fraction to lowest terms. $$\frac{-8}{48}$$

Step-by-Step Solution

Answer

The fraction \(-\frac{8}{48}\) reduces to \(-\frac{1}{6}\).

Step 1: Identify the Numbers

First, recognize that the fraction is \(-\frac{8}{48}\). This means we need to simplify the fraction \(\frac{8}{48}\) before considering the negative sign.

Step 2: Find the Greatest Common Divisor (GCD)

Identify the greatest common divisor (GCD) of the numerator 8 and the denominator 48. The divisors of 8 are 1, 2, 4, and 8. The divisors of 48 include 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. The largest common divisor is 8.

Step 3: Divide Both Numerator and Denominator by the GCD

Divide both the numerator and the denominator by their GCD, which is 8. So, \(\frac{8 \div 8}{48 \div 8} = \frac{1}{6}\).

Step 4: Apply the Negative Sign

Since the original fraction has a negative sign, apply it to the simplified fraction. Therefore, \(-\frac{8}{48} = -\frac{1}{6}\).

Key Concepts

Greatest Common Divisor

The greatest common divisor, or GCD, is essential in simplifying fractions. When reducing a fraction, identifying the GCD helps us find the largest number that divides both the numerator and the denominator without leaving a remainder. The GCD simplifies the fraction to its lowest terms, meaning the numerator and denominator share no common factors other than 1. To find the GCD:

• List all the divisors of the numerator and the denominator.
• Identify the largest common divisor shared by both numbers.

For example, in the fraction \(\frac{8}{48}\), the divisors of 8 are \(1, 2, 4, 8\) and the divisors of 48 are \(1, 2, 3, 4, 6, 8, 12, 16, 24, 48\). Here, the largest number present in both lists is 8, making it the GCD. This crucial step ensures that the resulting fraction is both true and as simplified as possible.

Negative Fractions

Negative fractions might seem tricky, but they follow simple rules. The negative sign can appear in the numerator, denominator, or right in front of the fraction's line, and it doesn’t affect the process of simplification. When simplifying a fraction like \(-\frac{8}{48}\), focus first on reducing the numeric values.Remember:

• The negative sign is not considered when finding the GCD.
• Once the fraction is simplified, reapply the negative sign to the resulting fraction.

For \(-\frac{8}{48}\), we ignore the negative sign until we've simplified \(\frac{8}{48}\) to \(\frac{1}{6}\). Then, simply apply the negative sign back to get \(-\frac{1}{6}\). Recognizing this rule helps when simplifying and understanding the meaning of negative fractions.

Reducing Fractions

Reducing fractions means simplifying them to their smallest possible version without changing their value. This is done by dividing the numerator and the denominator by their greatest common divisor (GCD). Once you understand how to find the GCD, the process becomes straightforward. Here's how you reduce a fraction:

• Find the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

For instance, to reduce \(\frac{8}{48}\), we see the GCD is 8. We divide the numerator and the denominator by 8, resulting in \(\frac{1}{6}\).Reducing fractions ensures clarity in math work and easier further calculations. It is a crucial skill, especially in comparing or adding fractions, where the simplest form helps in recognizing equivalences or calculating accurately.