Problem 83 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{18}{24}$$

Step-by-Step Solution

Answer

The fraction \(\frac{18}{24}\) simplifies to \(\frac{3}{4}\).

Step 1: Identify the Greatest Common Divisor (GCD)

To simplify a fraction, we need to find the greatest common divisor (GCD) of both the numerator and the denominator. For the fraction \(\frac{18}{24}\), start by listing the factors of 18 and 24. - The factors of 18 are: 1, 2, 3, 6, 9, 18.- The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.The common factors of 18 and 24 are: 1, 2, 3, 6. The greatest common factor is 6.

Step 2: Divide Both Numerator and Denominator by the GCD

Now that we have the GCD, simplify the fraction \(\frac{18}{24}\) by dividing both the numerator and the denominator by 6, the greatest common divisor.- Divide 18 by 6 to get 3.- Divide 24 by 6 to get 4.Thus, \(\frac{18}{24} = \frac{3}{4}\).

Step 3: Verify the Simplification

To ensure that \(\frac{3}{4}\) is in its lowest terms, check that the GCD of 3 and 4 is 1. The only common factor between 3 and 4 is 1, confirming that \(\frac{3}{4}\) cannot be simplified further and is the simplest form.

Key Concepts

Greatest Common Divisor

Finding the greatest common divisor (GCD) is the first step in simplifying fractions, and it's a key concept in fraction simplification.
The greatest common divisor of two numbers is the largest number that divides both of them without leaving a remainder.
To find the GCD of a fraction's numerator and denominator, follow these simple steps:

• List all factors for each number. Factors are numbers that multiply together to give the original number.
• Identify the common factors between the numerator and the denominator.
• Select the largest of these common factors; this number is your GCD.

By understanding and applying the concept of the greatest common divisor, we can simplify any fraction to its lowest terms.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form, or lowest terms.
A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1.
Here's how to approach fraction simplification:

• Use the GCD to divide both the numerator and the denominator of the fraction.
• Renumber the fraction after division. This means writing the new numerator over the new denominator.
• Verify the result is in its simplest form by checking that there are no common factors other than 1 left.

Simplifying fractions not only makes calculations easier, but it also provides clarity by presenting quantities in the most concise way possible.

Common Factors

Common factors are numbers that are factors of two or more numbers simultaneously.
When simplifying fractions, finding common factors is essential for determining the GCD.
Follow these steps to identify common factors:

• Start by listing out all factors for each number in the fraction.
• Look for numbers that appear in the factor lists of both the numerator and the denominator.
• The intersection of these lists gives you the common factors.

Recognizing common factors quickly leads to finding the greatest common divisor, thus making the fraction simplification process much more efficient.