Problem 21 - Elementary Algebra

Question

For the following problems, reduce, if possible, each fraction lowest terms. \(\frac{39}{13}\)

Step-by-Step Solution

Answer

Answer: The simplest form of the fraction \(\frac{39}{13}\) is 3.

Step 1: Find the GCD

To find the greatest common divisor (GCD) of 39 and 13, we notice that 13 is a factor of 39 (i.e., 39 is divisible by 13). So, the GCD is 13.

Step 2: Divide both numerator and denominator by the GCD

Since the GCD is 13, we'll divide both the numerator and the denominator by 13: $$ \frac{39}{13}=\frac{39\div13}{13\div13} $$

Step 3: Perform the division and simplify

Now, we perform the division and obtain the simplified fraction: $$ \frac{39\div13}{13\div13}=\frac{3}{1} $$ So, after reducing the given fraction to its lowest terms, we have \(\frac{39}{13} = \frac{3}{1} = 3\).

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is the largest positive integer that can exactly divide both the numerator and the denominator of a fraction without leaving a remainder. This concept is pivotal when simplifying fractions, as finding the GCD allows you to reduce the fraction to its simplest form.

• To find the GCD, you can list the factors of each number and identify the largest one that they share.
• Sometimes, it's easy to spot the GCD by recognizing that one number is a factor of another, just like in our exercise with 39 and 13.

Finding the GCD helps in ensuring our simplified fraction is as reduced as possible. It's a key step before dividing both the top and bottom numbers of a fraction.

Simplifying Fractions

Simplifying fractions involves reducing them to their lowest terms, meaning that the numerator and the denominator have only 1 as their common divisor. This makes the fraction easier to understand and use in calculations.

• Begin simplifying by finding the GCD of both the numerator and the denominator.
• Next, divide both the top and bottom parts of the fraction by the GCD.

For example, with the fraction \(\frac{39}{13}\), we first found the GCD to be 13. Then, dividing both 39 and 13 by 13, we arrived at \(\frac{3}{1}\), which is simply 3. Simplifying fractions is not only about simplifying numbers but also about making the fraction itself more manageable.

Numerator and Denominator

Every fraction consists of two main parts: the numerator and the denominator. Understanding these components is critical in various fraction operations including simplification and comparison.

• The numerator is the top number of a fraction, representing how many parts of the whole we have.
• The denominator is the bottom number, showing into how many equal parts the whole is divided.

In our example with \(\frac{39}{13}\), 39 is the numerator and 13 is the denominator. When we simplify the fraction, both these numbers need to be divided by their GCD, transforming them into smaller, more refined numbers that still represent the same value. Understanding the relationship between the numerator and the denominator is essential when working with fractions.