Problem 94 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{15}{40}$$

Step-by-Step Solution

Answer

The fraction \( \frac{15}{40} \) reduces to \( \frac{3}{8} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To reduce a fraction to its lowest terms, find the greatest common divisor (GCD) of the numerator and the denominator. For the fraction \( \frac{15}{40} \), list the factors of 15 and 40. The factors of 15 are \( 1, 3, 5, 15 \), and the factors of 40 are \( 1, 2, 4, 5, 8, 10, 20, 40 \). The greatest common factor is 5.

Step 2: Divide Both Numerator and Denominator

Divide the numerator and the denominator of the fraction \( \frac{15}{40} \) by their greatest common divisor, which is 5. \( \frac{15}{5} = 3 \) and \( \frac{40}{5} = 8 \). The fraction becomes \( \frac{3}{8} \).

Step 3: Verify Reduction

Confirm that the fraction \( \frac{3}{8} \) is in its simplest form by ensuring there are no common factors (other than 1) between the numerator 3 and the denominator 8. The factors of 3 are \( 1, 3 \), and the factors of 8 are \( 1, 2, 4, 8 \). They share no common factors other than 1.

Key Concepts

Understanding the Greatest Common Divisor

The concept of the Greatest Common Divisor (GCD) is essential when working to reduce fractions. It represents the largest positive integer that evenly divides two or more numbers without leaving a remainder. Finding the GCD of the numerator and denominator of a fraction allows you to reduce it to its simplest form.
To illustrate, let's consider the fraction \( \frac{15}{40} \).

• First, identify the factors of 15: \(1, 3, 5, 15\).
• Next, identify the factors of 40: \(1, 2, 4, 5, 8, 10, 20, 40\).
• The largest number common to both sets of factors is 5. Thus, the GCD of 15 and 40 is 5.

This number is key to simplifying the fraction because it represents the greatest amount by which you can divide both the numerator and the denominator to reduce the fraction cleanly.

Achieving the Simplest Form

Once you have the Greatest Common Divisor (GCD), you can use it to simplify the fraction. The simplest form of a fraction is when the numerator and denominator have no common factors other than 1.
Let's apply it to our example, \( \frac{15}{40} \), using the GCD we found, which is 5:

• Divide the numerator by the GCD: \( \frac{15}{5} = 3 \).
• Divide the denominator by the GCD: \( \frac{40}{5} = 8 \).

This simplifies the fraction to \( \frac{3}{8} \). Ensuring that there are no other common factors besides 1 confirms that this fraction is already in its simplest form. This process helps in making mathematical results easier to interpret and compare.

Clarifying Numerator and Denominator

In any fraction, understanding the roles of the numerator and denominator is crucial for grasping how fractions work. The numerator is the number above the fraction bar, representing how many parts we have or are considering. On the other hand, the denominator, the number below the fraction bar, signifies the total number of equal parts that make up a whole.
Take \( \frac{15}{40} \) as an example:

• The numerator is 15. This tells us we are considering 15 parts.
• The denominator is 40. This indicates that there are 40 equal parts in total.

As we reduce this fraction to \( \frac{3}{8} \), the changes reflect the same principle:

• The numerator becomes 3, meaning we now have 3 parts.
• The denominator becomes 8, which shows there are 8 parts making up a whole.

Understanding this basic structure is fundamental in manipulating and simplifying fractions effectively.