Problem 78 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{32}{28}$$

Step-by-Step Solution

Answer

The fraction \( \frac{32}{28} \) reduced to lowest terms is \( \frac{8}{7} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To reduce the fraction \( \frac{32}{28} \), we first need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 32 and 28 is 4 because 4 is the largest number that divides both 32 and 28 exactly without leaving a remainder.

Step 2: Divide Numerator and Denominator by the GCD

Once we have identified the GCD, we divide both the numerator and the denominator of the fraction by this GCD. So, we have to compute: \( \frac{32}{28} = \frac{32 \div 4}{28 \div 4} = \frac{8}{7} \).

Step 3: Check if the Fraction is in Lowest Terms

After dividing, we should check if the resulting fraction \( \frac{8}{7} \) can be reduced further. Since 8 and 7 do not have any common divisors other than 1, \( \frac{8}{7} \) is already in its simplest form.

Key Concepts

Greatest Common Divisor

Understanding the concept of the Greatest Common Divisor (GCD) is essential when reducing fractions. The GCD is the largest number that can exactly divide both the numerator and the denominator, leaving no remainder. For instance, to simplify the fraction \( \frac{32}{28} \), we first find their GCD. To do this:

• List the factors of 32, which are 1, 2, 4, 8, 16, and 32.
• List the factors of 28, which include 1, 2, 4, 7, 14, and 28.

By comparing these factors, we see that the largest number appearing in both lists is 4. Hence, the GCD of 32 and 28 is 4.
This knowledge helps us to efficiently simplify fractions by dividing both parts of a fraction by their GCD.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator have no common divisors other than 1. Once we have identified the greatest common divisor (GCD), simplifying becomes a straightforward process.To simplify \( \frac{32}{28} \), we:

• Divide the numerator (32) by the GCD (4) to get 8.
• Divide the denominator (28) by the GCD (4) to get 7.

This process results in the fraction \( \frac{8}{7} \).
Upon checking, if the numerator and denominator do not share any common factors besides 1, the fraction is in its simplest form.
Simplifying fractions is a fundamental math skill that not only helps in presenting answers more neatly but also aids in performing other operations, like addition or subtraction of fractions, more easily.

Numerator and Denominator

In any fraction, understanding the role of both the numerator and the denominator is crucial. The numerator is the top number in a fraction and it represents how many parts we have. The denominator is the bottom number and indicates the total number of equal parts into which the whole is divided.For example, in the fraction \( \frac{32}{28} \), 32 is the numerator while 28 is the denominator.
When simplifying a fraction, adjusting both the numerator and the denominator by the greatest common divisor (GCD) ensures accurate reduction of the fraction.

• A higher numerator compared to the denominator usually indicates a fraction greater than one.
• Fractions with the same numerator and denominator are equivalent to one.

By mastering these components, students can manipulate and understand fractions more intuitively, setting a solid foundation for more advanced mathematical concepts.