Problem 36 - Introductory Algebra for College Students

Question

Simplify each fraction by reducing it to its lowest terms. $$\frac{75}{80}$$

Step-by-Step Solution

Answer

The simplified form of \( \frac{75}{80} \) is \( \frac{15}{16} \).

Step 1: Identify the Numbers

In the given fraction \( \frac{75}{80} \), 75 is the numerator and 80 is the denominator.

Step 2: Find the Greatest Common Divisor (GCD) or Highest Common Factor (HCF)

The GCD or HCF is the largest number that divides into two or more numbers without a remainder. For the numbers 75 and 80, the GCD is 5.

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

Divide both 75 and 80 by the GCD, 5. This simplifies to \( \frac{75 \div 5}{80 \div 5} = \frac{15}{16} \).

Key Concepts

Greatest Common Divisor (GCD)

The greatest common divisor, also known as GCD, is crucial for simplifying fractions. It's the largest number that can divide each of the numbers in question without leaving a remainder. In the fraction \(\frac{75}{80}\), you need to find the GCD of 75 and 80. This determines how much both the numerator and the denominator can be reduced. Think of the GCD as the biggest number that can "go into" both given numbers evenly.

To find the GCD:

• List the factors of each number. Factors of 75 are 1, 3, 5, 15, 25, 75. Factors of 80 are 1, 2, 4, 5, 8, 10, 16, 20, 40, 80.
• Identify the largest factor they have in common. Here, both share the factor 5.

With this approach, you'll always find the greatest factor that can accurately bring down a fraction to its lowest possible terms.

Numerator and Denominator

Fractions consist of two parts—the numerator and the denominator. These are fundamental components that need to be clearly understood when working to simplify a fraction. The numerator is the top number in a fraction, representing how many parts of a whole you have. In \(\frac{75}{80}\), 75 is the numerator.

The denominator, on the other hand, is the bottom number in a fraction. It shows into how many equal parts the whole is divided. Here, 80 is the denominator. Understanding the roles and positions of the numerator and the denominator is essential when you apply any operation to a fraction, such as finding the GCD and simplifying.

Reducing Fractions

Reducing fractions, also known as simplifying fractions, is the process where you divide both the numerator and the denominator by their greatest common divisor. This results in the fraction being expressed in its simplest form. For example, in \(\frac{75}{80}\), both numbers can be divided by their GCD, 5.

This process requires a few simple steps:

• Find the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by this GCD.
• The result is a simplified fraction, making it easier to interpret and use in calculations.

Reducing fractions ensures they are as straightforward as possible.

Lowest Terms

When a fraction is expressed in its lowest terms, it means that the numerator and the denominator have no common factors other than 1. After reducing a fraction like \(\frac{75}{80}\) to \(\frac{15}{16}\), you achieve its lowest terms. This makes the fraction easier to work with and understand.

To ensure a fraction is in its lowest terms:

• Check if the numerator and the denominator have any common factors.
• If they only share the number 1, the fraction is already simplified.

Fractions in their lowest terms are efficient for mathematical operations and provide clarity and simplicity.