Problem 84 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{32}{40}$$

Step-by-Step Solution

Answer

The reduced fraction is \(\frac{4}{5}\).

Step 1: Identify the Greatest Common Divisor (GCD)

First, we need to find the greatest common divisor (GCD) of the numerator and the denominator, which are 32 and 40. To find the GCD, list the factors of each number. The factors of 32 are 1, 2, 4, 8, 16, 32. The factors of 40 are 1, 2, 4, 5, 8, 10, 20, 40. The largest common factor is 8.

Step 2: Divide the Numerator and Denominator by the GCD

To reduce the fraction to its lowest terms, we divide both the numerator and the denominator by their GCD. The GCD we identified is 8. Divide the numerator 32 by 8, which equals 4, and divide the denominator 40 by 8, which equals 5. The fraction thus simplifies to \(\frac{4}{5}\).

Step 3: Verify the Simplification

Finally, we need to ensure that the new fraction \(\frac{4}{5}\) is in its simplest form by checking if the numerator and denominator have any common divisors other than 1. The factors of 4 are 1, 2, 4, and the factors of 5 are 1, 5. Since 4 and 5 share no common factors other than 1, \(\frac{4}{5}\) is in its simplest form.

Key Concepts

Greatest Common Divisor

Understanding the concept of the Greatest Common Divisor (GCD) is crucial for fraction simplification. The GCD of two numbers is the largest positive integer that divides both numbers without leaving a remainder. For instance, when dealing with the fraction \( \frac{32}{40} \), we need to find the GCD to simplify the fraction properly.

To find the GCD, we first list all factors of each number involved:

• Factors of 32: 1, 2, 4, 8, 16, 32
• Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40

The shared numbers in these lists are the common factors, and the greatest of these is the GCD. In our example, you can see that 8 is the largest number present in both lists. Therefore, the GCD of 32 and 40 is 8, which we use in the next step to reduce the fraction.

Factors

Factors play an essential role in identifying the GCD and simplifying fractions. A factor of a number is a whole number that can be evenly divided into that number. Discovering the factors is quite straightforward: you test all numbers up to the original number to see which ones result in no remainder. Let's explore the factors for our example numbers:

• When identifying the factors of 32, you start from 1 and go up: 1, 2, 4, 8, 16, 32. Each number divides 32 without a remainder.
• Similarly, to find the factors of 40, start from 1 as well: 1, 2, 4, 5, 8, 10, 20, 40.

Listing these factors is an easy process and helps check which number, as the GCD, can simplify the original fraction. It's simple; just see which numbers appear in both lists! In our case, once again, that number is 8.

Simplest Form

Writing fractions in their simplest form serves to make calculations easier and numbers more understandable. To achieve this, a fraction is simplified when the numerator (top number) and denominator (bottom number) have no common factors other than 1.

Thus, after dividing both 32 and 40 by their GCD of 8, we receive the reduced fraction of \( \frac{4}{5} \). To ensure \( \frac{4}{5} \) is truly simplified, we need to confirm there are no additional common factors aside from 1:

• The factors of 4 are 1, 2, 4.
• The factors of 5 are 1, 5.

Since they share no factors other than 1, \( \frac{4}{5} \) is indeed in its simplest form. Simplifying fractions is invaluable in both mathematics and daily life calculations, as it provides clarity and ease of understanding.