Problem 36 - Elementary and Intermediate Algebra

Question

Simplify. $$ \frac{20}{26} $$

Step-by-Step Solution

Answer

The simplified form of \[ \frac{20}{26} \] is \[ \frac{10}{13} \].

Step 1: Find the Greatest Common Divisor (GCD)

First, determine the greatest common divisor (GCD) of 20 and 26. The GCD of two numbers is the largest number that divides both of them without leaving a remainder.

Step 2: Calculate the GCD

The factors of 20 are 1, 2, 4, 5, 10, and 20. The factors of 26 are 1, 2, 13, and 26. The common factors are 1 and 2, so the GCD is 2.

Step 3: Divide the Numerator and Denominator by the GCD

Divide both the numerator (20) and the denominator (26) by the GCD (2). This can be written as: \[ \frac{20 \div 2}{26 \div 2} = \frac{10}{13} \]

Step 4: Verify the Simplification

Double-check the result to ensure the fraction is in its simplest form. Since 10 and 13 have no common factors other than 1, the fraction \[ \frac{10}{13} \] cannot be simplified further.

Key Concepts

finding GCD

Finding the Greatest Common Divisor (GCD) is a crucial step in simplifying fractions. The GCD is the largest number that can evenly divide both the numerator and the denominator without leaving a remainder. For example, to find the GCD of 20 and 26, we first list out the factors of each number:
- Factors of 20: 1, 2, 4, 5, 10, 20
- Factors of 26: 1, 2, 13, 26
The common factors between 20 and 26 are 1 and 2. Of these, the largest is 2, so the GCD is 2.
Knowing how to find the GCD helps in simplifying fractions quickly and correctly. It's like finding a common building block between two numbers.

numerator and denominator

In a fraction, the numerator is the number above the line, and the denominator is the number below it. For instance, in the fraction \( \frac{20}{26} \), 20 is the numerator and 26 is the denominator.
Understanding the roles of the numerator and denominator is essential when working with fractions. The numerator represents how many parts we have, while the denominator indicates how many parts the whole is divided into.
When simplifying a fraction, we divide both the numerator and the denominator by their GCD. This process reduces the fraction to its simplest form, where both numbers are as small as possible and still retain the same value.

simplified form

Simplifying a fraction means reducing it to its simplest form. This involves finding the GCD of the numerator and the denominator, then dividing both by this number.
Using the fraction from our example, \( \frac{20}{26} \), we found the GCD to be 2. By dividing both the numerator and the denominator by 2, we get:
\[ \frac{20 \ div 2}{26 \ div 2} = \frac{10}{13} \]
The fraction \( \frac{10}{13} \) is in its simplified form because 10 and 13 have no common factors other than 1. This means we can't reduce the fraction any further.
Simplified fractions are easier to work with in mathematical operations and provide a clearer view of the proportion between the numerator and the denominator.