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Chapter 1: Problem 119

Simplify: $$\frac{264}{468}$$.

Short Answer

Expert verified
The simplified form of \( \frac{264}{468} \) is \( \frac{22}{39} \).

Step by step solution

01

- Identify the Greatest Common Divisor (GCD)

Find the greatest common divisor (GCD) of the numerator (264) and the denominator (468). To do this, list the factors of each number and identify the largest factor that they have in common.
02

- List Factors

For 264, the factors are: 1, 2, 3, 4, 6, 8, 11, 12, 22, 24, 33, 44, 66, 88, 132, 264. For 468, the factors are: 1, 2, 3, 4, 6, 12, 13, 18, 26, 36, 39, 52, 78, 117, 156, 234, 468.
03

- Determine the GCD

The common factors of 264 and 468 are: 1, 2, 3, 4, 6, 12. The greatest of these is 12. Therefore, the GCD of 264 and 468 is 12.
04

- Divide the Numerator and Denominator by the GCD

Divide both the numerator and the denominator by their GCD (12). This gives: \( \frac{264 \div 12}{468 \div 12} = \frac{22}{39} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Divisor
The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD is crucial in simplifying fractions.

To identify the GCD, list all the factors of the numerator and the denominator. Then, look for the largest factor they share.

For our example \( \frac{264}{468} \), the GCD of 264 and 468 is 12. Once you have the GCD, you can divide both the numerator and the denominator by this number to simplify the fraction.
Numerator and Denominator
Every fraction is made up of two parts: the numerator and the denominator.

The numerator is the top number and represents how many parts we have. In our example, the numerator is 264.

The denominator is the bottom number and shows into how many parts the whole is divided. In the example, the denominator is 468.

Simplifying the fraction involves making both these numbers smaller but equivalent in value. You do this by dividing both the numerator and the denominator by their GCD.
Factorization
Factorization is the process of breaking down a number into its prime factors. These are the prime numbers that multiply together to give the original number.

For 264, the prime factors are 2, 2, 2, 3, and 11. For 468, the prime factors are 2, 2, 3, 3, and 13.

Finding these prime factors helps simplify the fraction because it makes it easier to identify the common factors between the numerator and the denominator. By listing these common factors, we can find the GCD, which is essential for reducing the fraction to its simplest form.

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