Problem 13 - Prealgebra

Question

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. $$\frac{10}{12}$$

Step-by-Step Solution

Answer

The fraction \( \frac{10}{12} \) simplifies to \( \frac{5}{6} \).

Step 1: Identify the GCD

To simplify the fraction \( \frac{10}{12} \), first find the greatest common divisor (GCD) of the numerator 10 and the denominator 12. We determine the divisors: 10 is divisible by 1, 2, 5, and 10; 12 is divisible by 1, 2, 3, 4, 6, and 12. The largest common divisor is 2.

Step 2: Divide Numerator and Denominator by the GCD

Divide both the numerator and the denominator by their GCD. For \( \frac{10}{12} \), divide both 10 and 12 by 2: \( \frac{10 \div 2}{12 \div 2} = \frac{5}{6} \).

Step 3: Confirm the Simplification

Check if \( \frac{5}{6} \) can be simplified further. Since the greatest common divisor of 5 and 6 is 1, the fraction is already in its simplest form.

Key Concepts

Greatest Common Divisor (GCD)

The Greatest Common Divisor, often abbreviated as GCD, is a key concept when simplifying fractions. It refers to the greatest positive integer that divides both the numerator and the denominator without leaving a remainder. Understanding the GCD is crucial for simplifying fractions effectively.

To find the GCD, list all divisors of both numbers involved. Let's take the example of the fraction \( \frac{10}{12} \):

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

The largest number that is present in both lists is 2, making it the GCD. This number is the key to reducing the fraction to its simplest form. By dividing both the numerator and the denominator by the GCD, you simplify the fraction.

Numerator and Denominator

In any fraction, the part above the line is called the numerator. It represents how many parts of a whole you have. The part below the line is the denominator, which indicates how many parts the whole is divided into. Together, they create the fraction that represents a division.

• Numerator example: In the fraction \( \frac{10}{12} \), 10 is the numerator.
• Denominator example: In the same fraction, 12 is the denominator.

Proper understanding of these components allows you to manipulate fractions correctly. When simplifying, you aim to minimize the size of these numbers by using the GCD. Carefully dividing the numerator and the denominator by the GCD results in a simpler form of the fraction.

Simplest Form

A fraction is in its simplest form when no number other than 1 can divide both the numerator and the denominator evenly. Simplifying a fraction involves reducing it to this state by eliminating common factors.

For example, after finding the GCD of the fraction \( \frac{10}{12} \), we divided both the numerator and the denominator by 2:

• New Numerator: 10 divided by 2 equals 5.
• New Denominator: 12 divided by 2 equals 6.

This gives us the simplified fraction \( \frac{5}{6} \). To confirm it's in the simplest form, ensure that no other common divisor exists between the numerator and denominator, except 1. Since the GCD of 5 and 6 is 1, \( \frac{5}{6} \) cannot be simplified further. Therefore, it is in its simplest form.