Problem 8 - Elementary Algebra

Question

For exercises 1-12, simplify. $$ \frac{21}{63} $$

Step-by-Step Solution

Answer

\frac{1}{3}

Step 1 - Identify the Numbers

Look at the numerator (21) and the denominator (63) in the fraction \(\frac{21}{63}\).

Step 2 - Find the Greatest Common Divisor (GCD)

Determine the largest number that divides both 21 and 63 without leaving a remainder. The greatest common divisor of 21 and 63 is 21.

Step 3 - Divide the Numerator and Denominator by the GCD

Divide both the numerator and the denominator by their greatest common divisor. \(\frac{21 \/ 21}{63 \/ 21} = \frac{1}{3}\).

Step 4 - Write the Simplified Fraction

After dividing, the fraction simplifies to \(\frac{1}{3}\).

Key Concepts

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is the largest number that can be evenly divided into both the numerator and denominator of a fraction. For example, in the fraction \(\frac{21}{63}\), the GCD of 21 and 63 is found by identifying the largest common factor of both numbers. Here are a few steps to find the GCD:

• List the factors of both numbers. For 21, the factors are 1, 3, 7, and 21. For 63, the factors are 1, 3, 7, 9, 21, and 63.
• Identify the common factors. The common factors of 21 and 63 are 1, 3, 7, and 21.
• Select the largest common factor. The largest common factor is 21, so the GCD of 21 and 63 is 21.

Using the GCD, we can simplify the fraction by dividing both the numerator and the denominator by this number.

Numerator and Denominator

The numerator is the top number of a fraction, and the denominator is the bottom number. In the fraction \(\frac{21}{63}\), 21 is the numerator and 63 is the denominator. Here’s how these terms work:

• The numerator indicates how many parts of the whole are being considered. For instance, in \(\frac{21}{63}\), 21 parts of the whole are considered.
• The denominator indicates the total number of parts the whole is divided into. In this fraction, the whole is divided into 63 parts.

Understanding the roles of the numerator and denominator is essential when simplifying fractions, as it helps to correctly identify the parts of the fraction that need to be divided by the GCD.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form. This is done by dividing both the numerator and the denominator by their Greatest Common Divisor (GCD). Here's the step-by-step process for simplifying \(\frac{21}{63}\):

• Identify the numerator and denominator. For \(\frac{21}{63}\), the numerator is 21 and the denominator is 63.
• Find the GCD. The GCD of 21 and 63 is 21.
• Divide both the numerator and the denominator by the GCD. This means dividing 21 by 21 and 63 by 21.
• Simplify the fraction to \(\frac{1}{3}\) as both 21 divided by 21 equals 1, and 63 divided by 21 equals 3.

By following these steps, the fraction \(\frac{21}{63}\) simplifies to \(\frac{1}{3}\), which is its simplest form.