Problem 49 - Prealgebra

Question

Reduce each fraction to lowest terms. a. \(\frac{6}{51}\) b. \(\frac{6}{52}\) c. \(\frac{6}{54}\) d. \(\frac{6}{56}\) e. \(\frac{6}{57}\)

Step-by-Step Solution

Answer

a. \(\frac{2}{17}\), b. \(\frac{3}{26}\), c. \(\frac{1}{9}\), d. \(\frac{3}{28}\), e. \(\frac{2}{19}\).

Step 1: Simplify 6/51

Find the greatest common divisor (GCD) of 6 and 51. The factors of 6 are 1, 2, 3, 6, and the factors of 51 are 1, 3, 17, 51. The GCD is 3. Divide both numerator and denominator by 3: \( \frac{6}{51} = \frac{6 \div 3}{51 \div 3} = \frac{2}{17} \).

Step 2: Simplify 6/52

Find the GCD of 6 and 52. The factors of 6 are 1, 2, 3, 6, and the factors of 52 are 1, 2, 4, 13, 26, 52. The GCD is 2. Divide both the numerator and denominator by 2: \( \frac{6}{52} = \frac{6 \div 2}{52 \div 2} = \frac{3}{26} \).

Step 3: Simplify 6/54

Find the GCD of 6 and 54. The factors of 6 are 1, 2, 3, 6, and the factors of 54 are 1, 2, 3, 6, 9, 18, 27, 54. The GCD is 6. Divide both the numerator and denominator by 6: \( \frac{6}{54} = \frac{6 \div 6}{54 \div 6} = \frac{1}{9} \).

Step 4: Simplify 6/56

Find the GCD of 6 and 56. The factors of 6 are 1, 2, 3, 6, and the factors of 56 are 1, 2, 4, 7, 8, 14, 28, 56. The GCD is 2. Divide both the numerator and denominator by 2: \( \frac{6}{56} = \frac{6 \div 2}{56 \div 2} = \frac{3}{28} \).

Step 5: Simplify 6/57

Find the GCD of 6 and 57. The factors of 6 are 1, 2, 3, 6, and the factors of 57 are 1, 3, 19, 57. The GCD is 3. Divide both the numerator and denominator by 3: \( \frac{6}{57} = \frac{6 \div 3}{57 \div 3} = \frac{2}{19} \).

Key Concepts

Understanding the Greatest Common Divisor (GCD)

When simplifying fractions, finding the greatest common divisor, or GCD, is essential. It's the largest number that can evenly divide both the numerator and the denominator of a fraction. The process involves the following steps:

• List all factors of both the numerator and the denominator.
• Identify the common factors—those that appear in both lists.
• The highest of these common factors is the GCD.

For example, if we take the fraction \( \frac{6}{51} \), the factors of 6 are 1, 2, 3, and 6, while the factors of 51 are 1, 3, 17, and 51. The common factor is 3, making it the GCD. By dividing both 6 and 51 by 3, we simplify the fraction to \( \frac{2}{17} \).
Using the GCD helps streamline the simplification process and ensures that fractions are reduced accurately and efficiently.

The Importance of Numerator and Denominator

In fractions, the numerator is the top number, while the denominator is the bottom number. Together, they represent a part of a whole or a division of quantities.

• The numerator indicates how many parts we have.
• The denominator shows the total number of equal parts the whole is divided into.

To illustrate, take \( \frac{6}{52} \). Here, 6 is the numerator, meaning we have six parts, and 52 is the denominator, indicating the whole is divided into 52 equal parts. Understanding this division helps us see how much of a whole we have or are working with.
When simplifying, it's crucial to maintain this ratio correctly even after dividing both numbers by the GCD. For \( \frac{6}{52} \), dividing each by the GCD, which is 2, gives us \( \frac{3}{26} \), which still represents the same portion of the whole set.

How to Reduce Fractions Efficiently

Reducing fractions means simplifying them to their lowest terms, where the numerator and denominator have no common factors other than 1. This process makes calculations easier and results more comprehensible.
To reduce a fraction:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and denominator by the GCD.

Consider the fraction \( \frac{6}{54} \). The factors of 6 and 54 reveal that their GCD is 6. By dividing both the numerator and the denominator by 6, the fraction simplifies to \( \frac{1}{9} \).
Reducing fractions is a valuable skill not just for simplifying numbers, but also for enhancing understanding in math problems. This process ensures fractions are kept simple, neat, and ready for further use in mathematical operations or comparisons.