Problem 94 - Algebra 1: Concepts and Skills

Question

Write the fraction in simplest form. (Skills Review p. 763) $$ \frac{28}{35} $$

Step-by-Step Solution

Answer

The simplest form of the fraction \(\frac{28}{35}\) is \(\frac{4}{5}\).

Step 1: Find the Greatest Common Divisor (GCD)

First identify the greatest common divisor of 28 and 35. The divisors of 28 are: 1, 2, 4, 7, 14, 28. The divisors of 35 are: 1, 5, 7, 35. Therefore, the GCD of 28 and 35 is 7.

Step 2: Divide the Numerator and Denominator by the GCD

Once the GCD has been found, divide both the numerator and the denominator of the fraction by it. In this case, divide 28 and 35 by 7. This gives us: \(\frac{28}{7}=4\) and \(\frac{35}{7}=5\)

Step 3: Write the Simplified Fraction

After finding the results from step 2, it can be seen that the fraction \(\frac{28}{35}\) simplifies to \(\frac{4}{5}\). This is the simplest form.

Key Concepts

Greatest Common Divisor (GCD)

Understanding the Greatest Common Divisor (GCD), also known as the Greatest Common Factor (GCF), is critical when simplifying fractions. The GCD is the highest number that divides both the numerator and the denominator without leaving a remainder. To find it, list out all of the divisors of both numbers and look for the largest number that appears on both lists.

For instance, in the problem \(\frac{28}{35}\), the divisors of 28 are 1, 2, 4, 7, 14, and 28, while the divisors of 35 are 1, 5, 7, and 35. The largest number that appears in both lists is 7, making it the GCD of 28 and 35. By identifying and using the GCD, we can simplify the fraction to its simplest form efficiently.

Numerator and Denominator

In any fraction, the top number is called the numerator and the bottom number is called the denominator. The numerator represents the number of equal parts we have, while the denominator signifies the total number of equal parts that make up a whole. Simplifying a fraction means adjusting both the numerator and the denominator so that they are as small as possible, yet still represent the same value as the original fraction.

For example, with \(\frac{28}{35}\), the numerator is 28 and the denominator is 35. To simplify, we leverage the GCD of these numbers. Dividing both by their GCD, which is 7, gives us a new numerator of 4 and a new denominator of 5, resulting in the simplified fraction \(\frac{4}{5}\).

Simplest Form

A fraction is in its simplest form when the numerator and the denominator are both as small as possible and there are no common divisors between them except for 1. This means that they are in their lowest terms and the fraction cannot be reduced any further. Simplifying fractions is not just about making the numbers smaller; it's about representing the same quantity in a more basic and more easily understandable way.

The fraction \(\frac{28}{35}\), when simplified using the GCD, becomes \(\frac{4}{5}\). Here, 4 and 5 do not share any common divisors other than 1, meaning \(\frac{4}{5}\) is in its simplest form and conveys the same value as \(\frac{28}{35}\), but more succinctly.