Problem 32 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each fraction by reducing it to its lowest terms. $$\frac{18}{45}$$

Step-by-Step Solution

Answer

The simplified form of the fraction \(\frac{18}{45}\) is \(\frac{2}{5}\).

Step 1: Identification

Identify the given fraction which is \(\frac{18}{45}\)

Step 2: Find the Greatest Common Divisor (GCD)

Calculate the GCD of 18 and 45 which is 9.

Step 3: Divide by the GCD

Divide both the numerator and the denominator by the GCD. So, \(\frac{18}{9}\) becomes 2 and \(\frac{45}{9}\) becomes 5.

Step 4: Write the Reduced Fraction

The reduced form of the given fraction is \(\frac{2}{5}\).

Key Concepts

Greatest Common Divisor

When simplifying fractions, one important concept is the Greatest Common Divisor (GCD). The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. By finding the GCD, we can reduce fractions effectively, making them easier to work with in calculations. For instance, in the fraction \( \frac{18}{45} \), the GCD of 18 and 45 is 9. This means that 9 is the biggest number that can evenly divide both 18 and 45.

Finding the GCD typically involves listing the factors of each number and identifying the largest factor that is shared. Factors of 18 include 1, 2, 3, 6, 9, and 18, while factors of 45 include 1, 3, 5, 9, 15, and 45. The largest common factor, therefore, is 9. Understanding this concept helps greatly when moving onto the next step of reducing fractions.

Reducing Fractions

Reducing fractions might sound complex, but it's actually a straightforward process. Once you have the GCD, you can simplify or reduce a fraction by dividing both the numerator and the denominator by this number. This process maintains the value of the fraction while simplifying its appearance.

To reduce \( \frac{18}{45} \), you divide both 18 and 45 by their GCD, which is 9. So, you get \( \frac{18}{9} = 2 \) and \( \frac{45}{9} = 5 \). Therefore, the reduced fraction is \( \frac{2}{5} \). Reducing fractions is an essential skill because it can make further arithmetic simpler and reduce the potential for errors in calculations.

Numerator and Denominator

A fraction consists of two parts: the numerator and the denominator. The numerator is the top number and represents how many parts of the whole are being considered. The denominator is the bottom number and indicates the total number of equal parts the whole is divided into.

For example, in \( \frac{18}{45} \), 18 is the numerator and 45 is the denominator. During fraction simplification, focus on these two numbers. You aim to divide both by the GCD to modify the fraction's appearance without changing its value. This relationship between numerator and denominator is key to understanding how fractions work and their simplification process.

Lowest Terms

The objective of reducing fractions is to bring them to their lowest terms, meaning the fraction is simplified as much as possible. A fraction is in its lowest terms when the numerator and the denominator cannot be divided by any number other than 1.

When you reduce \( \frac{18}{45} \) using its GCD, you get \( \frac{2}{5} \), which is the fraction in its lowest terms. This form is the simplest representation and is often preferred for clarity and ease of calculation. Understanding lowest terms is useful in comparisons and operations with other fractions.