Problem 50 - Prealgebra

Question

Reduce each fraction to lowest terms. a. \(\frac{6}{42}\) b. \(\frac{6}{44}\) c. \(\frac{6}{45}\) d. \(\frac{6}{46}\) e. \(\frac{6}{48}\)

Step-by-Step Solution

Answer

a. \(\frac{1}{7}\), b. \(\frac{3}{22}\), c. \(\frac{2}{15}\), d. \(\frac{3}{23}\), e. \(\frac{1}{8}\).

Step 1: Understanding the Problem

The task is to simplify each given fraction \(\frac{a}{b}\) to its lowest terms by dividing both the numerator \(a\) and the denominator \(b\) by their greatest common divisor (GCD).

Step 2: Simplifying \(\frac{6}{42}\)

Find the GCD of 6 and 42, which is 6. Divide both the numerator and the denominator by 6. \(\frac{6 \div 6}{42 \div 6} = \frac{1}{7}\). So, \(\frac{6}{42}\) simplifies to \(\frac{1}{7}\).

Step 3: Simplifying \(\frac{6}{44}\)

Find the GCD of 6 and 44, which is 2. Divide both numbers by 2. \(\frac{6 \div 2}{44 \div 2} = \frac{3}{22}\). So, \(\frac{6}{44}\) simplifies to \(\frac{3}{22}\).

Step 4: Simplifying \(\frac{6}{45}\)

Find the GCD of 6 and 45, which is 3. Divide both numbers by 3. \(\frac{6 \div 3}{45 \div 3} = \frac{2}{15}\). So, \(\frac{6}{45}\) simplifies to \(\frac{2}{15}\).

Step 5: Simplifying \(\frac{6}{46}\)

Find the GCD of 6 and 46, which is 2. Divide both numbers by 2. \(\frac{6 \div 2}{46 \div 2} = \frac{3}{23}\). So, \(\frac{6}{46}\) simplifies to \(\frac{3}{23}\).

Step 6: Simplifying \(\frac{6}{48}\)

Find the GCD of 6 and 48, which is 6. Divide both numbers by 6. \(\frac{6 \div 6}{48 \div 6} = \frac{1}{8}\). So, \(\frac{6}{48}\) simplifies to \(\frac{1}{8}\).

Key Concepts

Understanding the Greatest Common Divisor (GCD)

The concept of the Greatest Common Divisor, or GCD, is a key element to understanding how to simplify fractions. The GCD of two numbers is the largest number that can evenly divide both of them.
For example, when you want to simplify the fraction \(\frac{6}{42}\), you look for the largest number that can divide both 6 and 42 without leaving a remainder.
To find the GCD, list out the factors of each number:

• Factors of 6: 1, 2, 3, 6
• Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

The largest common factor is 6, thus the GCD is 6.
When you have the GCD, you can then divide both the numerator and the denominator of the fraction by this number to simplify the fraction.

Simplifying Fractions to Lowest Terms

Simplifying a fraction to its lowest terms involves reducing it so that the numerator and the denominator have no common factors other than 1. This is where the GCD becomes crucial.
Let's take the fraction \(\frac{6}{44}\) as another example. The GCD here is 2.
By dividing both the numerator and the denominator by 2, you reduce the fraction:

• Numerator: \(6 \div 2 = 3\)
• Denominator: \(44 \div 2 = 22\)

Thus, the fraction \(\frac{6}{44}\) simplifies to \(\frac{3}{22}\).
By always checking for and dividing by the GCD, any fraction can be reduced to its lowest terms.

Understanding Numerators and Denominators

In any fraction \(\frac{a}{b}\), the numerator is the top number \(a\), and the denominator is the bottom number \(b\). These elements define how many parts the fraction is divided into and how many parts are being considered.
Let's consider the fraction \(\frac{6}{45}\). Here,

• Numerator (\(a\)): 6
• Denominator (\(b\)): 45

By dividing both by their GCD of 3, the fraction is simplified. This brings us to:

• New numerator: \(6 \div 3 = 2\)
• New denominator: \(45 \div 3 = 15\)

Thus, \(\frac{6}{45}\) is simplified to \(\frac{2}{15}\).
Simplification helps in better understanding and comparing fractions, as it presents them in their most concise form.