Problem 79 - Algebra 1: Concepts and Skills

Question

\(\frac{6}{20}\)

Step-by-Step Solution

Answer

\(\frac{3}{10}\)

Step 1: Identify the Greatest Common Divisor

Finding the greatest common divisor (GCD) means identifying the biggest number that evenly divides into both 6 and 20. In this case, it is clear that the GCD is 2.

Step 2: Divide by the Greatest Common Divisor

Divide both the numerator (6) and the denominator (20) by the GCD (2). This yields \(\frac{6}{2} / \frac{20}{2}\)

Step 3: Simplify the Fraction

Calculate the values of the numerator and denominator. This will give the final, simplified fraction, \(\frac{6/2}{20/2} = \frac{3}{10}\)

Key Concepts

Greatest Common Divisor

The Greatest Common Divisor, often abbreviated as GCD, is a crucial concept in simplifying fractions. It is the largest positive integer that divides two or more numbers without leaving a remainder. For example, in the fraction \(\frac{6}{20}\), we look for a number that can evenly divide both 6 and 20. This number is 2, since both 6 and 20 can be divided by 2 without any remainder.
Understanding how to find the GCD is essential:

• List the divisors (whole numbers) of each number.
• Identify the biggest divisor that appears in both lists.

Using the GCD to simplify fractions not only reduces the size of the numbers but also makes them easier to work with in calculations.

Numerator

The numerator is a key component of a fraction, sitting on top of the fraction line. It tells us how many parts of a whole are being considered. In our original fraction \(\frac{6}{20}\), the numerator is 6. This means that out of the 20 equal parts making up a whole, 6 parts are being used or counted.
When simplifying fractions, it's important to divide the numerator by the greatest common divisor. For example, by dividing the original numerator 6 by the GCD 2, we get 3. This gives us the simplified numerator for our fraction \(\frac{3}{10}\).
Understanding the role of the numerator helps us to accurately represent parts of a whole in a simplified form.

Denominator

The denominator is found at the bottom of a fraction, below the fraction line. It represents the total number of equal parts that make up a whole. In \(\frac{6}{20}\), the denominator is 20, indicating that the whole is divided into 20 equal parts.
When simplifying a fraction, we also need to divide the denominator by the GCD. For instance, dividing 20 by 2 (the GCD we found earlier) gives us 10, which becomes the denominator in the simplified fraction \(\frac{3}{10}\).
By reducing the denominator, we ensure that the fraction is expressed in its simplest form, making it easier to interpret and use in further mathematical calculations.