Problem 23 - Algebra A Combined Function

Question

Simplify each fraction. See Examples 8 through \(10 .\) $$ \frac{18}{30} $$

Step-by-Step Solution

Answer

The simplified fraction is \( \frac{3}{5} \).

Step 1: Find the Greatest Common Divisor (GCD)

To simplify a fraction, start by finding the greatest common divisor (GCD) of the numerator and the denominator. Here, the numbers are 18 and 30. The divisors of 18 are: 1, 2, 3, 6, 9, 18; the divisors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30. The greatest common divisor is 6.

Step 2: Divide Numerator and Denominator by the GCD

Now divide both the numerator and the denominator by their greatest common divisor, 6. \( \frac{18}{30} = \frac{18 ÷ 6}{30 ÷ 6} = \frac{3}{5} \).

Step 3: Simplify the Resulting Fraction

After dividing, check if \( \frac{3}{5} \) can be simplified further. In this case, 3 and 5 have no common divisors other than 1, so \( \frac{3}{5} \) is in its simplest form.

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is a key concept when simplifying fractions. It refers to the largest number that can evenly divide both the numerator and the denominator without leaving a remainder. For instance, to simplify a fraction like \( \frac{18}{30} \), you start by listing the divisors for each number:

• The divisors of 18 are: 1, 2, 3, 6, 9, 18
• The divisors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30

From these lists, the greatest common divisor is 6, as it is the largest number dividing both 18 and 30 perfectly. Finding the GCD is essential because it helps you determine by what number to divide both the numerator and the denominator to simplify the fraction.

Numerator and Denominator

Understanding the roles of the numerator and denominator is crucial in working with fractions. The numerator is the top part of a fraction, representing how many parts of a whole are being considered. The denominator, on the other hand, is the bottom part, showing the total number of equal parts the whole is divided into.For example, in \( \frac{18}{30} \), 18 is the numerator indicating 18 parts out of the 30 parts which make up a whole. To simplify the fraction, both these numbers must be divided by their greatest common divisor (GCD), which is 6 in this case.So, \( \frac{18}{30} \) transforms to \( \frac{3}{5} \) after dividing:

• Divide 18 (numerator) by 6 to get 3.
• Divide 30 (denominator) by 6 to get 5.

This process helps in reducing the fraction to its simplest form.

Simplest Form

The simplest form of a fraction is when the numerator and the denominator have no common divisors other than 1. This means the fraction cannot be reduced any further.After dividing the numerator and denominator by their greatest common divisor (GCD), as we did with \( \frac{18}{30} \), you get \( \frac{3}{5} \). Now, 3 and 5 are coprime numbers; they share no common factors other than 1. This confirms that \( \frac{3}{5} \) is indeed in its simplest form.Working towards the simplest form is important in mathematics because:

• It provides a more intuitive understanding of the fraction.
• It allows easier comparison with other fractions.
• It helps in performing further operations like addition or subtraction of fractions.

Always check for common divisors at the end of simplifying to ensure the fraction is fully reduced.