Problem 10 - Elementary Algebra

Question

For the following problems, reduce, if possible, each fraction lowest terms. \(\frac{32}{28}\)

Step-by-Step Solution

Answer

Question: Simplify the fraction \(\frac{32}{28}\) to its lowest terms. Answer: \(\frac{8}{7}\)

Step 1: Find the Greatest Common Divisor (GCD)

To simplify the fraction, we first need to find the GCD of both numerator (32) and the denominator (28). The GCD is the largest number that divides both numbers without leaving a remainder. Using prime factorization: \(32 = 2^5\) \(28 = 2^2 \cdot 7\) The common factors are \(2^2\), so the GCD is \(2^2 = 4\).

Step 2: Divide Numerator and Denominator by GCD

Now that we have found the GCD (4), divide both the numerator (32) and the denominator (28) by this number: \(\frac{32}{4} = 8\) \(\frac{28}{4} = 7\)

Step 3: Write the Simplified Fraction

Now that we have divided both the numerator and denominator by the GCD, we can write the fraction in its simplified form: \(\frac{32}{28} = \frac{8}{7}\) The simplified fraction is \(\frac{8}{7}\).

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is a vital concept when simplifying fractions. It refers to the largest positive integer that exactly divides two or more integers without leaving any remainder. To simplify a fraction, finding the GCD of its numerator and denominator is the first crucial step.

When calculating the GCD, it helps to keep in mind that:

• All numbers are divisible by 1 and themselves, but only the largest common factor is the GCD.
• The GCD cannot be larger than the smallest of the numbers being compared.
• It's the key to reducing fractions to their simplest form.

In our example, we calculated the GCD of 32 and 28 by using their prime factorization, discovering that the GCD is 4. By dividing both the numerator and the denominator by this number, we successfully simplified the fraction.

Prime Factorization

Prime factorization is a brilliant method used to break down numbers into their basic building blocks, which are prime numbers. Prime numbers are numbers greater than 1 that have no factors other than 1 and themselves. For instance, the numbers like 2, 3, 5, 7, etc., are prime numbers. They play a huge role in determining the GCD of two numbers.

To use prime factorization:

• Decompose each number into its prime factors.
• Find the common prime factors between the numbers.
• Multiply these common factors to find the GCD.

Take 32 and 28:

• 32 can be factored into prime numbers as \(2^5\).
• 28 can be factored as \(2^2 \cdot 7\).
• Both numbers share the prime factor \(2^2\), so the GCD is \(2^2 = 4\).

This made it possible to simplify the fraction easily by eliminating these common factors.

Lowest Terms

Simplifying fractions to their lowest terms means reducing them so that the numerator and denominator have no common factors other than 1. This version of the fraction is considered its simplest form. The process becomes straightforward once you have the GCD at hand.

Here’s how you achieve this:

• Find the GCD of the fraction's numerator and denominator.
• Divide both the numerator and the denominator by this GCD.
• The result is the fraction in its lowest terms.

In our specific example, we started with the fraction \(\frac{32}{28}\). By dividing both 32 and 28 by their GCD of 4, we were left with \(\frac{8}{7}\). Now, \(\frac{8}{7}\) has no common factors besides 1, ensuring it is indeed in its lowest terms.