Problem 19 - Elementary Algebra

Question

For the following problems, reduce, if possible, each fraction lowest terms. \(\frac{27}{81}\)

Step-by-Step Solution

Answer

Answer: The reduced form of the fraction \(\frac{27}{81}\) is \(\frac{1}{3}\).

Step 1: Identify the given fraction

The given fraction is \(\frac{27}{81}\).

Step 2: Find the GCD of the numerator and the denominator

The numerator is 27 and the denominator is 81. We need to find the GCD of 27 and 81. First, let's list the factors of the two numbers: Factors of 27: 1, 3, 9, 27 Factors of 81: 1, 3, 9, 27, 81 As we can see, the greatest common divisor (GCD) of 27 and 81 is 27.

Step 3: Divide the numerator and the denominator by the GCD

Now, we will divide both the numerator and the denominator by the GCD (27). Numerator: \(\frac{27}{27} = 1\) Denominator: \(\frac{81}{27} = 3\)

Step 4: Write the reduced fraction

So, the reduced fraction is \(\frac{1}{3}\). Therefore, the fraction \(\frac{27}{81}\) can be reduced to \(\frac{1}{3}\).

Key Concepts

Greatest Common Divisor (GCD)

Understanding the Greatest Common Divisor (GCD) is key to simplifying fractions effectively. The GCD is the largest positive integer that divides two or more numbers without leaving a remainder. When reducing a fraction, finding the GCD of its numerator and denominator allows us to simplify the fraction to its simplest form.

To determine the GCD, follow these steps:

• List all positive factors of the numerator and the denominator.
• Identify the largest factor that appears in both lists. This is the GCD.

For example, with the fraction \(\frac{27}{81}\), the factors of 27 are 1, 3, 9, and 27, and the factors of 81 include 1, 3, 9, 27, and 81. The largest common factor from these lists is 27, making it the GCD. Knowing the GCD helps us to divide both numerator and denominator, simplifying the fraction with confidence.

Factors

Factors are numbers that divide another number evenly, meaning without leaving a remainder. Each number has its own set of factors, which can be useful for various mathematical operations, including finding the GCD.

When simplifying fractions, identifying factors is crucial as it helps find common divisors between the numerator and denominator. Let's consider the process:

• Begin by listing all the factors of the numerator.
• Do the same for the denominator.

With our fraction \(\frac{27}{81}\), the factors of 27 are 1, 3, 9, and 27, and for 81, the factors are 1, 3, 9, 27, and 81. These factor lists facilitate the identification of the GCD, which in turn assists in simplifying the fraction correctly.

Numerator and Denominator

In a fraction, the numerator and the denominator are essential components that need to be understood before simplifying. The numerator is the top number, representing the number of parts we have. The denominator is the bottom number, indicating the total number of equal parts into which something is divided.

In the fraction \(\frac{27}{81}\), 27 is the numerator, and 81 is the denominator. To simplify the fraction:

• Divide both the numerator and the denominator by their GCD, which in this case is 27.
• This results in a simplified fraction: \(\frac{1}{3}\).

Remember, the goal of simplifying is to express the same value with smaller values for the numerator and the denominator, making it easier to understand and use in further calculations.