Problem 98 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{121}{132}$$

Step-by-Step Solution

Answer

The reduced form of \(\frac{121}{132}\) is \(\frac{11}{12}\).

Step 1: Identify the Greatest Common Divisor (GCD)

The first step is to find the greatest common divisor of the numerator and the denominator of the fraction \(\frac{121}{132}\). Begin by determining the factors of both numbers. The factors of 121 are 1, 11, and 121, as 121 is a perfect square of 11. The factors of 132 are 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, and 132. The greatest common factor they share is 11.

Step 2: Divide the Numerator and Denominator by the GCD

Using the GCD identified from the previous step, divide both the numerator (121) and the denominator (132) by 11 to simplify the fraction. So, we perform the division: \(\frac{121}{11} = 11\) and \(\frac{132}{11} = 12\). This simplifies the fraction to \(\frac{11}{12}\).

Step 3: Verify the Simplification

Check if the fraction can be reduced further by examining the new numerator (11) and denominator (12). Since 11 is a prime number and does not divide 12, the fraction \(\frac{11}{12}\) is already in its simplest form.

Key Concepts

Greatest Common Divisor

The Greatest Common Divisor (GCD) is a critical concept in simplifying fractions. It’s the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD helps us reduce fractions to their simplest form.

To determine the GCD of two numbers, follow these steps:

• List down all the factors of each number.
• Identify the largest factor that both numbers share.

In the fraction \(\frac{121}{132}\), we first determine the factors of each:
- **Factors of 121**: 1, 11, 121
- **Factors of 132**: 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132

The common factors are 1 and 11, where 11 is the GCD.

Once you've identified the GCD, you can simplify the fraction by dividing both the numerator and the denominator by this number. This method is a reliable way to ensure that the fraction is in the simplest form.

Reducing Fractions

Reducing fractions involves simplifying them so that the numerator and denominator have no common divisors, apart from 1. This makes the fraction easier to understand and work with.

To reduce a fraction:

• First, identify the GCD of the numerator and denominator.
• Divide both the top and bottom of the fraction by the GCD.

In our example of \(\frac{121}{132}\), once we identified the GCD as 11, we divided both numbers by 11:

\(\frac{121}{11} = 11\) and \(\frac{132}{11} = 12\).

This simplifies the fraction to \(\frac{11}{12}\).

After performing this division, check that no further simplification is possible by ensuring that the numerator and denominator do not have any other common factors besides 1.

Prime Numbers

Prime numbers play a significant role in mathematics, especially in simplifying fractions. A prime number is greater than 1 and has no divisors other than 1 and itself.

Prime numbers are useful in identifying factors. They can also aid in confirming when a fraction is in its simplest form.

In the case of \(\frac{11}{12}\), 11 is a prime number. This means it only divides by 1 and 11. Since 11 cannot divide 12, \(\frac{11}{12}\) remains in its simplest form.

Recognizing prime numbers can thus help quickly verify simplification:

• If the numerator or denominator is prime and doesn't divide the other number, the fraction is fully reduced.

Understanding prime numbers and their properties is valuable for reducing fractions confidently and efficiently.