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Chapter 1: Problem 45

Simplify. $$ \frac{17}{51} $$

Short Answer

Expert verified
The simplified form of \(\frac{17}{51}\) is \(\frac{1}{3}\).

Step by step solution

01

- Identify the Greatest Common Divisor (GCD)

To simplify the fraction \(\frac{17}{51}\), first identify the greatest common divisor (GCD) of the numerator and the denominator. Check if both numbers have any common factors. In this case, the factors of 17 are 1 and 17, and the factors of 51 are 1, 3, 17, and 51.
02

- Determine the Common Factor

Notice that both 17 and 51 share a common factor of 17. This is the largest number that divides both the numerator and the denominator without leaving a remainder.
03

- Divide the Numerator and Denominator by the GCD

Now, divide both the numerator and the denominator by their greatest common divisor (17): \(\frac{17 \div 17}{51 \div 17}\). This simplifies to \(\frac{1}{3}\).

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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, finding the greatest common divisor (GCD) is a crucial step. The GCD is the largest number that can exactly divide both the numerator and denominator.
To find the GCD:
1. List all factors of the numerator and denominator.
2. Identify the largest factor that both numbers have in common.
For example, with \( \frac{17}{51} \), we find the factors of 17 are 1 and 17, and the factors of 51 are 1, 3, 17, and 51. The common factor is 17. Therefore, the GCD is 17.
numerator
The numerator is the top part of a fraction. It tells us how many parts of the whole we are considering. In \( \frac{17}{51} \), 17 is the numerator.
When simplifying fractions, the numerator plays a key role in determining how the fraction can be reduced. By understanding and finding the GCD with the denominator, you can divide the numerator by this common divisor to simplify the fraction.
In our example, 17 is divided by 17 (the GCD), resulting in the simplified numerator: 1.
denominator
The denominator is the bottom part of a fraction. It indicates the total number of equal parts the whole is divided into. In the fraction \( \frac{17}{51} \), 51 is the denominator.
To simplify a fraction, we need to understand the role of the denominator and work with the GCD. Just like with the numerator, we divide the denominator by the GCD.
In our example, dividing 51 by the GCD (which is 17), we get the simplified denominator: 3.
common factors
Common factors are numbers that divide exactly into two or more numbers. To simplify a fraction, we identify common factors of both the numerator and the denominator.
Here's how to find common factors:
1. List all the factors of the numerator.
2. List all the factors of the denominator.
3. Identify the numbers that appear in both lists.
For \( \frac{17}{51} \), the factors of 17 are 1 and 17, and the factors of 51 are 1, 3, 17, and 51. The common factor is 17, allowing us to divide both terms by 17 to simplify the fraction to \( \frac{1}{3} \).

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