Problem 81 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{12}{18}$$

Step-by-Step Solution

Answer

The fraction \( \frac{12}{18} \) simplifies to \( \frac{2}{3} \).

Step 1: Determine the Greatest Common Divisor (GCD)

To simplify the fraction \( \frac{12}{18} \), we need to find the greatest common divisor (GCD) of the numerator (12) and the denominator (18). The divisors of 12 are 1, 2, 3, 4, 6, and 12. The divisors of 18 are 1, 2, 3, 6, 9, and 18. Comparing these, the greatest common divisor is 6.

Step 2: Divide Numerator and Denominator by the GCD

Using the GCD (which is 6), divide both the numerator and the denominator to simplify the fraction. \( \frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3} \).

Step 3: Verify the Simplified Fraction

Check if the simplified fraction \( \frac{2}{3} \) can be reduced any further. The only common divisor of 2 and 3 is 1, which means this is the simplest form of the fraction.

Key Concepts

Greatest Common Divisor

The concept of the Greatest Common Divisor (GCD) is pivotal in simplifying fractions. When we talk about GCD, we mean the largest number that can exactly divide both the numerator and the denominator of a fraction without leaving a remainder. For instance, consider the fraction \( \frac{12}{18} \). To simplify this, we first need to find their GCD.

To find the GCD, let's list the divisors:

• Divisors of 12 include: 1, 2, 3, 4, 6, 12
• Divisors of 18 include: 1, 2, 3, 6, 9, 18

These lists show us that both numbers share several divisors. The largest of these shared numbers is 6. Hence, the GCD of 12 and 18 is 6. Finding the GCD is a crucial step, as it guides us in reducing the fraction to its simplest form.

Simplifying Fractions

When it comes to fractions, simplifying them means making the fraction as simple as possible by ensuring both the numerator and the denominator are only divisible by 1. Let's continue with our example of \( \frac{12}{18} \).

Once we've determined that the GCD is 6, we can simplify the fraction by dividing both the numerator and the denominator by this number.

Performing the division, we have:

• \( 12 \div 6 = 2 \)
• \( 18 \div 6 = 3 \)

This gives us the fraction \( \frac{2}{3} \). Simplifying fractions helps in making calculations easier, and ensures clarity. It confirms that the simplified fraction is at its "lowest terms."

Once simplified, check if any further reduction is possible. However, if the only common divisor is 1, then the fraction is already in its simplest form.

Numerator and Denominator

Understanding what the numerator and the denominator represent is fundamental for working with fractions. A fraction typically stands for a part of a whole, and it is expressed as \( \frac{a}{b} \), where \( a \) is the numerator and \( b \) is the denominator.

In our example of \( \frac{12}{18} \), 12 is the numerator, and 18 is the denominator. The numerator tells us how many parts we are interested in, while the denominator indicates into how many equal parts the whole is divided. This foundational concept ensures clarity when performing operations with fractions.

When simplifying, we look to reduce these with respect to their greatest common divisor. Comprehending how numerators and denominators interact allows for easy manipulation and simplification of fractions, a skill that is highly useful in many math problems and real-world applications.