Problem 136 - Introductory and Intermediate Algebra for College Students 4th

Question

Explain how to reduce a fraction to its lowest terms. Give an example with your explanation.

Step-by-Step Solution

Answer

To reduce a fraction to its lowest term, find the greatest common divisor of the numerator and denominator, and divide both by this number. For example, \( \frac{16}{24} \) reduces to \( \frac{2}{3} \), as 8 is the GCD of 16 and 24.

Step 1: Identify the Fraction

Look at the fraction that needs to be reduced. For example, let's take \( \frac{16}{24} \).

Step 2: Identify the Greatest Common Divisor (GCD)

The GCD of two numbers is the biggest number that can divide both of them without remainder. One way to find this is to list the factors of each number. For \(16: 1, 2, 4, 8, 16\) and for \(24: 1, 2, 3, 4, 6, 8, 12, 24\). The largest number found in both lists is \(8\). So, \(8\) is the greatest common divisor.

Step 3: Divide Both the Numerator and Denominator by the GCD

This is how we reduce the fraction to its lowest terms. Divide both the numerator (16) and denominator (24) by the GCD (8): \( \frac{16}{8} / \frac{24}{8} \)= \( \frac{2}{3} \). So \( \frac{16}{24}\) reduces to \( \frac{2}{3} \).

Key Concepts

Greatest Common Divisor

Finding the greatest common divisor (GCD) is crucial when reducing fractions to their lowest terms. But what exactly is the GCD? The GCD of two numbers is the largest number that can divide both numbers without leaving a remainder.
For example, let's consider the numbers 16 and 24. We need to find all the numbers that can divide each of them completely.

• For 16, the divisors are: 1, 2, 4, 8, and 16.
• For 24, the divisors are: 1, 2, 3, 4, 6, 8, 12, and 24.

By comparing these lists, we see that 8 is the largest number that appears in both, making it the greatest common divisor. Once we identify the GCD, we can use it to simplify fractions by dividing both the numerator and the denominator by this number.

Numerator and Denominator

Before you can reduce a fraction, it's important to understand its components: the numerator and the denominator. The numerator is the top part of a fraction, while the denominator is the bottom part.
For example, in the fraction \(\frac{16}{24}\), 16 is the numerator, and 24 is the denominator. The numerator tells us how many parts we have, and the denominator tells us how many equal parts make up a whole.
Knowing these parts is helpful because when you reduce a fraction, you divide both the numerator and the denominator by the same number, ensuring the value of the fraction does not change. This maintains the mathematical balance as you simplify the fraction.

Factorization

Factorization is a method that helps us break down or decompose numbers into their factors. This technique is quite useful when simplifying fractions.

To factor a number, you think of it as a product of its prime numbers. For instance, let's factor 16 and 24:

• For 16: The factors are 2, 2, 2, and 2 (Note that \(2^4 = 16\)).
• For 24: The factors are 2, 2, 2, and 3 (Note that \(2^3 \times 3 = 24\)).

Since we're looking for the largest factor that is common to both lists (as seen in the GCD), this process ensures accurate results when reducing fractions. Factorization not only leads us to the GCD but also reinforces why dividing the numerator and denominator by this number will simplify the fraction.