Problem 18 - Basic College Mathematics

Question

Simplify. $$ \frac{8}{10} $$

Step-by-Step Solution

Answer

\( \frac{4}{5} \)

Step 1 - Find the Greatest Common Divisor (GCD)

Identify the greatest common divisor (GCD) of the numerator (8) and the denominator (10). The GCD is the largest number that evenly divides both numbers. The GCD of 8 and 10 is 2.

Step 2 - Divide the Numerator and Denominator by the GCD

Divide both the numerator and the denominator by the GCD (2). \(\frac{8 \/ 2}{10 \/ 2} \) results in \(\frac{4}{5} \).

Step 3 - Simplify

The fraction cannot be simplified any further. So, \(\frac{4}{5} \) is the simplified form of the given fraction.

Key Concepts

Greatest Common Divisor (GCD)

The Greatest Common Divisor, often abbreviated as GCD, is a fundamental concept in mathematics. It refers to the largest number that can evenly divide two or more numbers without leaving a remainder. To find the GCD, we need to identify all the common factors of the given numbers and select the largest one.
For example, let's determine the GCD of 8 and 10, which are the numerator and denominator in the fraction \(\frac{8}{10}\). The factors of 8 are 1, 2, 4, and 8. The factors of 10 are 1, 2, 5, and 10. The common factors are 1 and 2, with 2 being the largest. Therefore, the GCD of 8 and 10 is 2.
When working with fractions, finding the GCD allows us to simplify the fraction effectively.

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form. This makes it easier to understand and work with. To simplify a fraction, we divide both the numerator (the top number) and the denominator (the bottom number) by their Greatest Common Divisor (GCD).
Let's apply this to \(\frac{8}{10}\). We first identified the GCD as 2. Next, we divide both the numerator and the denominator by 2: \(\frac{8 ÷ 2}{10 ÷ 2}\). This gives us \(\frac{4}{5}\), which is the simplified form of \(\frac{8}{10}\).
This process ensures that the numerator and denominator are as small as possible, while still retaining the same value.

Numerator and Denominator

In any fraction, there are two key parts: the numerator and the denominator. Understanding these parts is crucial to working with fractions.
The numerator is the top number of a fraction and represents the number of parts we have. For instance, in the fraction \(\frac{8}{10}\), 8 is the numerator.
The denominator is the bottom number of a fraction and signifies the total number of equal parts the whole is divided into. In the fraction \(\frac{8}{10}\), 10 is the denominator.
By dividing both the numerator and the denominator by their GCD, we can reduce the fraction to its simplest form. For example, dividing 8 and 10 each by 2, we get \(\frac{4}{5}\). This makes the fraction easier to work with and understand.