Problem 79 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{36}{10}$$

Step-by-Step Solution

Answer

The reduced fraction is \( \frac{18}{5} \).

Step 1: Find the Greatest Common Divisor (GCD)

Identify the greatest common divisor of the numerator and the denominator. In this case, we need to find the GCD of 36 and 10. The divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36, and the divisors of 10 are 1, 2, 5, 10. The largest common divisor is 2.

Step 2: Divide Both Terms by the GCD

To simplify the fraction, divide both the numerator and denominator by the GCD found in the previous step. Divide 36 by 2 to get 18, and divide 10 by 2 to get 5. Thus, \( \frac{36}{10} \) becomes \( \frac{18}{5} \).

Step 3: Confirm the Fraction is in Lowest Terms

Check if the new fraction can be simplified further by finding the GCD of 18 and 5. Since their only common divisor is 1, the fraction \( \frac{18}{5} \) is in its simplest form.

Key Concepts

Greatest Common Divisor

Understanding the concept of the Greatest Common Divisor (GCD) is essential for simplifying fractions. The GCD of two numbers is the largest number that can perfectly divide both without leaving a remainder. It's like finding the biggest chunk that evenly fits into both your numerator and denominator.

To find the GCD, list all the divisors of the numerator and the denominator. From our example with the fraction \(\frac{36}{10}\):

• Divisors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
• Divisors of 10: 1, 2, 5, 10

Looking at both lists, the largest common number is 2. This makes 2 our GCD. By using the GCD, you can simplify your fraction efficiently, which leads us to the next concept.

Numerator and Denominator

A fraction is made up of two parts: the numerator and the denominator. The numerator is the top number that represents how many parts we have. The denominator is the bottom number that shows how many parts make a whole. For example, in the fraction \(\frac{36}{10}\), 36 is the numerator and 10 is the denominator.

The relationship between these two numbers is crucial when simplifying fractions. By dividing both the numerator and the denominator by their greatest common divisor, you reduce the fraction to a simpler form without changing its value. Continuing from our example, divide both 36 and 10 by their GCD, which is 2:

• 36 divided by 2 equals 18
• 10 divided by 2 equals 5

This gives us a simplified fraction: \(\frac{18}{5}\). Remember, when simplifying, you maintain the proportionate relationship between the numerator and the denominator.

Lowest Terms

To express a fraction in its lowest terms means reducing it as much as possible, ensuring no number other than 1 can divide both the numerator and denominator evenly. This process results in a fraction where the numerator and denominator are co-prime, meaning their only common divisor is 1.

Returning to our simplified fraction \(\frac{18}{5}\), we confirm it is in the lowest terms by checking the GCD of 18 and 5. Since their only common divisor is 1, we cannot reduce \(\frac{18}{5}\) any further. It is essential to ensure your fractions are in the lowest terms for clarity and ease of understanding, especially when working with equations or comparing fractions.

• Simplifying with the GCD helps achieve this quickly.
• Reducing fractions enhances their simplicity without altering the actual value.

When checking for lowest terms, just remember: if you've used the GCD correctly, you're already there!