Problem 69 - Prealgebra

Question

Reduce to lowest terms. $$\frac{75}{100}$$

Step-by-Step Solution

Answer

The lowest terms of \( \frac{75}{100} \) is \( \frac{3}{4} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To reduce \( \frac{75}{100} \) to its lowest terms, first identify the greatest common divisor (GCD) of the numerator (75) and the denominator (100). Both 75 and 100 are divisible by 25, which is their GCD.

Step 2: Divide the Numerator and Denominator by the GCD

Divide both the numerator and the denominator by their GCD, which is 25: \( \frac{75}{100} = \frac{75 \div 25}{100 \div 25} \).

Step 3: Simplify the Fraction

After performing the division, simplify the fraction: \( \frac{75 \div 25}{100 \div 25} = \frac{3}{4} \).

Key Concepts

Greatest Common Divisor

The Greatest Common Divisor, or GCD for short, is a fundamental concept when it comes to simplifying fractions. Think of it as the largest number that can evenly divide both the numerator and the denominator of a fraction, without leaving any remainder.
To find the GCD, you can list all the factors of both numbers and identify the largest one they share. Another efficient approach is the Euclidean algorithm, a step-by-step method involving division and remainders. For example, to find the GCD of 75 and 100, one might:

• Recognize that 75's factors are 1, 3, 5, 15, 25, and 75.
• And 100's factors are 1, 2, 4, 5, 10, 20, 25, 50, and 100.
• The common factors are 1, 5, and 25, with the largest being 25.

Thus, 25 is the GCD of 75 and 100, making it key to simplifying their fraction.

Simplifying Fractions

Simplifying fractions is all about making the numbers smaller and easier to work with, without changing their value. You achieve this by dividing both the numerator and the denominator by their Greatest Common Divisor (GCD).
Let's use the fraction \(\frac{75}{100}\) as an example again. We've established that their GCD is 25. Simplifying the fraction involves dividing both by this GCD:

• Divide the numerator: \(75 \div 25 = 3\)
• Divide the denominator: \(100 \div 25 = 4\)

So, \(\frac{75}{100}\) becomes \(\frac{3}{4}\) after simplification. This means the fraction represents the same value but in a form that's cleaner and more understandable.Simplifying fractions not only makes them look nicer but also makes mathematical calculations simpler and more intuitive.

Numerator and Denominator

The concepts of numerator and denominator are essential for understanding fractions. A fraction is composed of two parts:- **Numerator**: This is the top number of a fraction. It shows how many parts of a whole we are considering.- **Denominator**: This is the bottom number. It indicates into how many equal parts the whole is divided.For example, in the fraction \(\frac{75}{100}\):

• 75 is the numerator. It tells us the count of parts being taken.
• 100 is the denominator. It represents the total number of equal parts the whole is divided into.

Understanding the roles of the numerator and denominator helps in operations like adding, subtracting, or simplifying fractions. Remember:

• If you change the numerator, you change how many parts of the whole you're considering.
• If you change the denominator, you change the size of each part of the whole.

This distinction is vital when simplifying fractions because you have to ensure you're dividing both by the same non-zero number to maintain the fraction's value.