Problem 37 - Elementary and Intermediate Algebra

Question

Simplify. $$ \frac{16}{56} $$

Step-by-Step Solution

Answer

\(\frac{2}{7}\)

Step 1: Identify the Greatest Common Divisor (GCD)

To simplify the fraction \(\frac{16}{56}\), start by finding the greatest common divisor (GCD) of the numerator (16) and the denominator (56). The GCD is the largest number that divides both 16 and 56 without leaving a remainder.

Step 2: Divide the Numerator and Denominator by the GCD

The GCD of 16 and 56 is 8. Now, divide both the numerator and the denominator by this number: \(\frac{16 \div 8}{56 \div 8} = \frac{2}{7}\).

Step 3: Write the Simplified Fraction

The fraction has been simplified. Therefore, \(\frac{16}{56} = \frac{2}{7}\).

Key Concepts

Greatest Common Divisor

The Greatest Common Divisor (GCD) is a key concept in simplifying fractions. The GCD of two numbers is the largest number that can divide both of them without leaving a remainder. Finding the GCD helps you reduce the fraction to its simplest form.

For example, to simplify \(\frac{16}{56}\), you first need to find the GCD of 16 and 56. This means you look for the largest number that can evenly divide both 16 and 56. In this case, it is 8.

Numerator

The numerator is the top number of a fraction. In a fraction \(\frac{a}{b}\), 'a' is the numerator. It represents the number of parts you have out of the whole.

For instance, in the given exercise \(\frac{16}{56}\), 16 is the numerator. It tells you how many parts of the whole (which is represented by the denominator) are being considered in the fraction.

Denominator

The denominator is the bottom number of a fraction. In a fraction \(\frac{a}{b}\), 'b' is the denominator. It represents the total number of equal parts that make up the whole.

In the fraction \(\frac{16}{56}\), 56 is the denominator. It indicates the total number of parts into which the numerator's portions are divided.

Fraction Simplification

Fraction simplification is the process of making a fraction as simple as possible. You do this by dividing the numerator and the denominator by their GCD.

In the exercise, to simplify \(\frac{16}{56}\):

• First, find the GCD of 16 and 56, which is 8.
• Next, divide both the numerator and the denominator by the GCD.
• This results in \(\frac{16 \div 8}{56 \div 8} = \frac{2}{7}\).

So, the simplified form of the fraction is \(\frac{2}{7}\).