Problem 5 - Prealgebra

Question

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. $$\frac{64}{68}$$

Step-by-Step Solution

Answer

\( \frac{16}{17} \)

Step 1: Identify the Greatest Common Divisor (GCD)

To simplify the fraction \( \frac{64}{68} \), first find the greatest common divisor (GCD) of the numerator and denominator. List the factors of 64: 1, 2, 4, 8, 16, 32, 64, and the factors of 68: 1, 2, 4, 17, 34, 68. The largest common factor is 4.

Step 2: Divide Both Numerator and Denominator by the GCD

With the GCD found to be 4, divide both the numerator and the denominator by the GCD: \( \frac{64}{4} = 16 \) and \( \frac{68}{4} = 17 \). Therefore, the simplified form of \( \frac{64}{68} \) is \( \frac{16}{17} \).

Step 3: Verify Simplification

Verify that \( \frac{16}{17} \) is in its simplest form. Check the factors of 16: 1, 2, 4, 8, 16, and the factors of 17: 1, 17. Since 16 and 17 have no common factors other than 1, the fraction is in its simplest form.

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is a vital concept in simplifying fractions. It's the largest number that divides both the numerator and the denominator without leaving a remainder.
To determine the GCD, you need to find and compare the factors of both numbers involved in the fraction.
For instance, in the fraction \( \frac{64}{68} \), we list the factors of both 64 and 68:

• Factors of 64: 1, 2, 4, 8, 16, 32, 64
• Factors of 68: 1, 2, 4, 17, 34, 68

The greatest number they share as a factor is 4, making it the GCD. By using the GCD, you ensure that the fraction is simplified optimally and correctly.

Factors

Understanding factors is key to grasping how to simplify fractions. Factors are numbers that divide another number completely, without leaving a remainder. Finding all the factors of a number involves dividing that number by integers to see which divisions yield whole numbers.
Each number has a set of factors which are crucial when determining the greatest common divisor.
Take for example the number 64. Its divisors are integers where 64 divided by them leaves no leftover. Similarly, for 68, we find its set of factors. These factors help us find the GCD, which in turn assists the simplification of fractions.
Recognizing these factors not only aids in fraction simplification but also enhances overall number sense.

Numerator and Denominator

Fractions consist of two main parts: the numerator and the denominator. The numerator is the number on top, representing how many parts we have. The denominator, located below the line, indicates how many parts make up a whole.
For example, in the fraction \( \frac{64}{68} \), 64 is the numerator, and 68 is the denominator. These terms are essential because the process of simplification relies heavily on understanding their roles.

• The numerator tells how many portions are being considered.
• The denominator tells the total number of equal parts the whole is divided into.

Understanding these roles aids in the simplification process, where both numbers are divided by their greatest common divisor to reach their simplest form.

Simplified Form

The simplified form of a fraction is when the numerator and denominator no longer share any common factors other than 1. Achieving this form is crucial for clarity and easy comparison of fractions.
Simplifying a fraction involves dividing both parts by the greatest common divisor. For example, dividing the numerator and denominator of \( \frac{64}{68} \) by 4 results in \( \frac{16}{17} \).
Now, 16 and 17 share no common factors except for 1, making \( \frac{16}{17} \) the simplest form. Simplified fractions are typically easier to work with in equations and comparisons. They also represent the fraction in its most reduced form, enhancing clarity in communication.