Problem 68 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{18}{14}$$

Step-by-Step Solution

Answer

The fraction \( \frac{18}{14} \) reduces to \( \frac{9}{7} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To simplify the fraction \( \frac{18}{14} \), we need to find the greatest common divisor (GCD) of the numerator (18) and the denominator (14). The GCD is the largest number that divides both numbers without leaving a remainder.

Step 2: Find the GCD

List the factors of both numbers. The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 14 are 1, 2, 7, and 14. The common factors are 1 and 2. The greatest of these is 2, so the GCD of 18 and 14 is 2.

Step 3: Divide the Numerator and Denominator by the GCD

Simplify the fraction by dividing both the numerator and the denominator by their GCD (2). This gives: \( \frac{18}{14} = \frac{18 \div 2}{14 \div 2} = \frac{9}{7} \).

Step 4: Verify the Simplified Fraction is in Lowest Terms

Check if \( \frac{9}{7} \) can be simplified further. The factors of 9 are 1, 3, and 9, while the factors of 7 are 1 and 7. Since they have no common factors other than 1, \( \frac{9}{7} \) is in its simplest form.

Key Concepts

Greatest Common Divisor (GCD)

Simplifying fractions is a straightforward process when you understand the concept of the Greatest Common Divisor (GCD). The GCD is the largest number that can divide both the numerator and the denominator of a fraction without leaving a remainder. This is key because it directly helps us reduce fractions to their lowest terms, making calculations simpler. To find the GCD of two numbers:

• List out all factors of each number.
• Identify the common factors between the two.
• The largest common factor is the GCD.

Taking our example, the GCD of 18 and 14 is 2. Using the GCD, we simplify the fraction \(\frac{18}{14}\) to \(\frac{9}{7}\). This new fraction is in its simplest form because 9 and 7 have no other common factors besides 1.

Numerator and Denominator

When dealing with fractions, the terms numerator and denominator are fundamental. Each fraction comprises these two parts where the numerator is the number above the fraction line, and the denominator is the number below.

• The numerator represents how many parts we have out of the whole.
• The denominator shows the total number of parts that make up a whole.

For the fraction \(\frac{18}{14}\), 18 is the numerator and 14 is the denominator. To simplify a fraction, we divide both the numerator and the denominator by their GCD. It’s essential to perform these operations correctly to ensure the fraction is reduced as much as possible, resulting in the same value but a simpler form.

Factors of Numbers

Understanding how to find factors of numbers is vital in simplifying fractions. Factors are the numbers you multiply together to get another number. Every number has a unique set of factors. To discover the factors of a number:

• Start by testing division with smaller numbers to see if there is no remainder.
• Continue with each subsequent number to check divisibility until you reach the number itself.

For example, the factors of 18 include 1, 2, 3, 6, 9, and 18. The factors of 14 are 1, 2, 7, and 14. The process of finding these factors helps in determining the GCD, which in turn allows us to simplify fractions effectively. Knowing these factors and identifying any common ones between two numbers is the basis for reducing fractions to their lowest terms.