Problem 82 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{18}{27}$$

Step-by-Step Solution

Answer

The fraction reduces to \( \frac{2}{3} \).

Step 1: Understanding the Problem

We need to simplify the fraction \( \frac{18}{27} \) by reducing it to its lowest terms. This involves finding a common divisor of the numerator and the denominator.

Step 2: Finding the Greatest Common Divisor (GCD)

First, we need to find the greatest common divisor of 18 and 27. By evaluating the factors, we find: - Factors of 18: 1, 2, 3, 6, 9, 18 - Factors of 27: 1, 3, 9, 27 The common factors are 1, 3, and 9. Thus, the greatest common divisor is 9.

Step 3: Dividing Both Terms by the GCD

We simplify \( \frac{18}{27} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 9:\[\frac{18 \div 9}{27 \div 9} = \frac{2}{3}\]

Step 4: Result Verification

Ensure the fraction \( \frac{2}{3} \) is in its simplest form. Since 2 and 3 have no common factors other than 1, \( \frac{2}{3} \) is indeed in the lowest terms.

Key Concepts

Greatest Common Divisor

When working with fractions, a crucial step in simplifying is finding the greatest common divisor (GCD). The GCD is the largest number that divides two or more numbers without leaving a remainder. Understanding how to find the GCD is key to simplifying fractions correctly.
For example, when we look at the fraction \( \frac{18}{27} \), first, we identify the factors of each number:

• Factors of 18: 1, 2, 3, 6, 9, 18
• Factors of 27: 1, 3, 9, 27

The common factors are the numbers that appear in both lists: 1, 3, and 9. Since the largest of these is 9, the GCD of 18 and 27 is 9.
Finding the GCD helps because it tells us what number we can divide both the numerator and the denominator by to simplify the fraction.

Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple as possible by dividing both the numerator and the denominator by their greatest common divisor.
For \( \frac{18}{27} \), once we determine that the GCD is 9, the next step is dividing both the numerator (18) and the denominator (27) by this GCD:

• Numerator: \( 18 \div 9 = 2 \)
• Denominator: \( 27 \div 9 = 3 \)

Thus, \( \frac{18}{27} \) simplifies to \( \frac{2}{3} \).
This process reduces the fraction to a simpler form without changing its value, making calculations smoother and more understandable.

Lowest Terms

Once a fraction is simplified, it is in its lowest terms. A fraction is in its lowest terms when the numerator and the denominator have no common factors other than 1.
After simplifying \( \frac{18}{27} \) to \( \frac{2}{3} \), check that 2 and 3 only share the factor 1. Since they do, \( \frac{2}{3} \) is in its lowest terms.
Having a fraction in its lowest terms is beneficial because it provides the simplest version of the value it represents. This makes further operations, like addition or multiplication with other fractions, easier and clearer.