Problem 10 - Fundamentals of Mathematics

Question

Reduce, if possible, each fraction to lowest terms. \((\) Section 4.4\() \frac{15}{51}\)

Step-by-Step Solution

Answer

The fraction in lowest terms is \( \frac{5}{17} \).

Step 1: Find the Greatest Common Divisor (GCD)

To simplify the fraction, we need to find the greatest common divisor of the numerator and the denominator. We start with 15, which factors to 3 × 5, and 51, which factors to 3 × 17. The common factor is 3.

Step 2: Divide the Numerator and Denominator by the GCD

Using the GCD we found, divide both the numerator and the denominator by 3. This process gives us the fraction \( \frac{15 \div 3}{51 \div 3} = \frac{5}{17} \).

Step 3: Verify the Simplified Fraction

Check if 5 and 17 have any common factors other than 1. Since they do not, \( \frac{5}{17} \) is the fraction in its lowest terms.

Key Concepts

Greatest Common Divisor

The greatest common divisor, often abbreviated as GCD, is a key concept in simplifying fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD of two numbers helps simplify a fraction to its simplest form.
To find the GCD, you can list the factors of each number and pick the largest common factor. For example:

• For the number 15, the factors are 1, 3, 5, and 15.
• For the number 51, the factors are 1, 3, 17, and 51.

The common factor here is 3, the largest number that divides both 15 and 51 evenly. This GCD is crucial because dividing both the numerator and the denominator by 3 will reduce the fraction to a simpler form.

Simplifying Fractions

Simplifying fractions is the process of reducing them to the simplest possible form without changing their value. This means rewriting the fraction so that the numerator and the denominator have no common factors other than 1. The key to simplifying is using the GCD.
Here's how you simplify a fraction using the GCD:

• Identify the GCD of the numerator and denominator.
• Divide the numerator by this GCD.
• Divide the denominator by this GCD.

For example, with the fraction \(\frac{15}{51}\), we found the GCD to be 3. Dividing both 15 and 51 by 3 transforms the fraction into \(\frac{5}{17}\). This indicates the original fraction has been scaled down to its simplest form.

Lowest Terms

When a fraction is in its lowest terms, it means the numerator and the denominator share no common factors except 1. Achieving the lowest terms makes the fraction as simple as possible, often enhancing clarity in mathematical operations.
How do you know if a fraction is in its lowest terms? By ensuring:

• You have divided the original numerator and denominator by their GCD.
• There are no further common factors left.

In the example of \(\frac{5}{17}\), checking factors reveals that 5 and 17 have no common divisors aside from 1. Since no other factors exist, this fraction is confirmed to be in its lowest terms, making it as straightforward as possible for further calculations or comparisons.