Problem 103 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{40}{18}$$

Step-by-Step Solution

Answer

The fraction \( \frac{40}{18} \) reduces to \( \frac{20}{9} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To reduce a fraction to its lowest terms, determine the greatest common divisor (GCD) of the numerator and the denominator. The numerator is 40, and the denominator is 18. Let's find the GCD of 40 and 18 by listing the factors. Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40 Factors of 18: 1, 2, 3, 6, 9, 18 The common factors are 1 and 2. Thus, the GCD is 2.

Step 2: Divide Both the Numerator and Denominator by the GCD

Once the GCD is identified, divide both the numerator and the denominator of the fraction by this number. Starting with the fraction \( \frac{40}{18} \), divide both the numerator and denominator by 2:\[\frac{40 \div 2}{18 \div 2} = \frac{20}{9}\]

Step 3: Verify the Simplified Fraction

After performing the division, verify whether \( \frac{20}{9} \) is in its simplest form. List factors of 20 and 9 to verify:Factors of 20: 1, 2, 4, 5, 10, 20 Factors of 9: 1, 3, 9There are no common factors other than 1, confirming that \( \frac{20}{9} \) is indeed in its simplest form.

Key Concepts

Greatest Common Divisor

The greatest common divisor, often abbreviated as GCD, is a vital concept when it comes to simplifying fractions. This is because the GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD is the first step to simplify fractions as it helps us reduce them to their lowest terms.

To find the GCD between two numbers:

• List all the factors of each number.
• Identify the common factors shared by the numbers.
• Select the largest common factor; this is the GCD.

For instance, with the fraction \( \frac{40}{18} \): - The factors of 40 include 1, 2, 4, 5, 8, 10, 20, and 40.- The factors of 18 include 1, 2, 3, 6, 9, and 18.- The common factors are 1 and 2.- Thus, the GCD is 2.By using the GCD, you can then simplify any fraction by dividing the numerator and the denominator by this number.

Numerator and Denominator

When working with fractions, it is crucial to understand the roles of the numerator and the denominator. The numerator is the top part of the fraction and indicates how many parts of the whole or collection are being considered. The denominator, meanwhile, is the bottom part and shows the total number of equal parts that make up the whole or collection.

For example, in the fraction \( \frac{40}{18} \):

• 40 is the numerator, indicating 40 parts out of a whole.
• 18 is the denominator, representing that the whole is divided into 18 equal parts.

Understanding their roles helps in both operations and reductions of fractions. When simplifying, the objective is to see if both the numerator and the denominator can be divided by the greatest common divisor to yield a simpler and equivalent fraction, like simplifying \( \frac{40}{18} \) to \( \frac{20}{9} \).

Factors of Numbers

Factors are numbers that divide another number completely, without leaving a remainder. For instance, when determining the simplest form of a fraction, it's helpful to know both the numerator's and the denominator's factors. These factors are crucial in finding the greatest common divisor (GCD), which is used to simplify fractions.

Consider the fraction \( \frac{40}{18} \):

• The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40.
• The factors of 18 are 1, 2, 3, 6, 9, and 18.

By comparing these factors, we can see that the greatest common factor for both numbers is 2. Factors not only help in reducing fractions but also play a key role in various other mathematical operations such as finding least common multiples, solving equations, and understanding number patterns.