Problem 46 - Algebra 1: Concepts and Skills

Question

Write the fraction in simplest form. $$ \frac{9}{27} $$

Step-by-Step Solution

Answer

Therefore, the simplest form of the fraction \( \frac{9}{27} \) is \( \frac{1}{3} \).

Step 1: Identify Numerator and Denominator

In the fraction \( \frac{9}{27} \), 9 is the numerator and 27 is the denominator.

Step 2: Determine the Greatest Common Divisor

Find the greatest common divisor (gcd) of 9 and 27. It turns out that 9 is the gcd as it is the highest number that both 9 and 27 can be evenly divided by.

Step 3: Divide the Numerator and Denominator by GCD

Divide both 9(numerator) and 27(denominator) by the gcd, 9. \(\frac{9}{9}\) gives 1 and \(\frac{27}{9}\) gives 3.

Key Concepts

Greatest Common Divisor

The greatest common divisor, often abbreviated as gcd, is a key concept when simplifying fractions. It is the largest number that can evenly divide both the numerator and the denominator of a fraction. Finding the gcd helps you know by how much you can shrink both the numerator and the denominator while maintaining the same value of the fraction. To find the gcd, you can list out all the divisors of both numbers and select the largest common one.

• For instance, with 9 and 27, their divisors are:
• 9: 1, 3, 9
• 27: 1, 3, 9, 27

The greatest common factor of these two lists is 9, which means 9 is the gcd. This gcd is used to simplify the fraction by dividing both numbers by it.

Numerator

In any fraction, the numerator is the top number. It represents how many parts of the whole are being considered. In our example, the fraction is \( \frac{9}{27} \), and the numerator is 9. Understanding the role of the numerator is essential as it provides the context on what you're dealing with in terms of parts or shares of a whole. During simplification, since you're trying to find an equivalent fraction that is simpler, you divide the numerator by the gcd.

• In the fraction \( \frac{9}{27} \), dividing the numerator 9 by the gcd 9 gives you 1.

This new numerator forms part of the simplified fraction, which is \( \frac{1}{3} \).

Denominator

The denominator is the number located below the line in a fraction. It indicates the total number of equal parts that make up a whole. In our fraction \( \frac{9}{27} \), the denominator is 27. The denominator shows how large the whole is in terms of specific units or parts. When simplifying, the denominator is divided by the gcd as well to find its simplest form.

• For the fraction \( \frac{9}{27} \), dividing 27 by the gcd, 9, results in 3.

Understanding the denominator also helps you understand the scale of the fraction and its components. In a simplified state, this turns our original fraction into \( \frac{1}{3} \). Each part is three times larger than those in the original fraction.