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Chapter 1: Problem 48

Simplify each fraction, if possible. $$ \frac{22}{77} $$

Short Answer

Expert verified
\(\frac{22}{77}\) simplifies to \(\frac{2}{7}\).

Step by step solution

01

Find the Greatest Common Divisor

First, identify the greatest common divisor (GCD) of the numerator and the denominator of the fraction \( \frac{22}{77} \). The factors of 22 are 1, 2, 11, and 22. The factors of 77 are 1, 7, 11, and 77. The common factors are 1 and 11, so the GCD is 11.
02

Divide Both Terms by the GCD

Divide both the numerator and the denominator by the greatest common divisor (GCD) found in Step 1. Divide 22 by 11 to get 2, and divide 77 by 11 to get 7. Thus, \( \frac{22}{77} \) simplifies to \( \frac{2}{7} \).
03

Verify the Simplification

After simplifying, check that \( \frac{2}{7} \) has no common factors other than 1. The factors of 2 are 1 and 2, and the factors of 7 are 1 and 7. The only common factor is 1, so \( \frac{2}{7} \) is indeed simplified.

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Key Concepts

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

Greatest Common Divisor
When simplifying fractions, a crucial step is identifying the greatest common divisor (GCD). The GCD is the largest number that can divide both the numerator and the denominator without leaving a remainder. This helps us in breaking down the fraction to its simplest form.

Let's consider the fraction \(\frac{22}{77}\). The GCD can be identified by first finding the factors of each number in the fraction. Factors are numbers that can divide a given number evenly.
  • The factors of 22 are 1, 2, 11, and 22.
  • The factors of 77 are 1, 7, 11, and 77.
Looking at both sets of factors, we see that the common ones are 1 and 11. Among these, 11 is the greatest. Therefore, the GCD of 22 and 77 is 11, which will help us simplify the fraction efficiently.
Factors
Factors play a significant role in the simplification of fractions. Factors are numbers that multiply together to produce another number. Identifying them correctly is essential in determining the greatest common divisor.

When simplifying \(\frac{22}{77}\), we list out all the possible factors for each part of the fraction. For example:
  • 22: 1, 2, 11, 22
  • 77: 1, 7, 11, 77
The common factors between these numbers are what we focus on to reduce the fraction. In this instance, we have the numbers 1 and 11 that fit as factors for both 22 and 77. Out of these, 11 is the greatest, so using it to divide both terms of the fraction will lead to a simplified form.
Simplified Fractions
Once we've found the greatest common divisor (GCD), we use it to simplify the fraction by dividing both the numerator and the denominator by the GCD.

For example, in the fraction \(\frac{22}{77}\):
  • Divide 22 (numerator) by 11 (GCD) to get 2.
  • Divide 77 (denominator) by 11 (GCD) to get 7.
After this division, \(\frac{22}{77}\) simplifies to \(\frac{2}{7}\). We then need to verify this new fraction to ensure it's in the most reduced form. We check the factors of 2 and 7:
  • 2: 1, 2
  • 7: 1, 7
The only common factor is 1, confirming that \(\frac{2}{7}\) is indeed the simplest form of the original fraction.

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