Problem 93 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{15}{16}$$

Step-by-Step Solution

Answer

The fraction \( \frac{15}{16} \) is already in its lowest terms.

Step 1: Understand the Problem

We need to reduce the fraction \( \frac{15}{16} \) to its lowest terms. This means we should find the greatest common divisor (GCD) of the numerator (15) and the denominator (16) and divide both by it.

Step 2: Find the GCD of 15 and 16

List the factors of 15 (1, 3, 5, 15) and the factors of 16 (1, 2, 4, 8, 16). The common factor is 1, which is the greatest common divisor.

Step 3: Divide Numerator and Denominator by the GCD

Since the GCD is 1, dividing both the numerator and the denominator by 1 keeps the fraction the same.

Step 4: Write the Fraction in Lowest Terms

The fraction \( \frac{15}{16} \) cannot be reduced further since its GCD is 1. Therefore, \( \frac{15}{16} \) is already in its lowest terms.

Key Concepts

greatest common divisor

Reducing fractions to their simplest form often involves finding the greatest common divisor (GCD). This is a crucial step in simplifying fractions.
The GCD is the largest number that divides both the numerator and the denominator without leaving any remainder.
To find it, list all the factors of both numbers.

• For example, to find the GCD of 15 and 16, identify their factors:
• Factors of 15 are 1, 3, 5, and 15.
• Factors of 16 are 1, 2, 4, 8, and 16.

The highest common factor between these two sets is 1.
This means that 1 is their GCD, which indicates that \(\frac{15}{16}\) is already in its simplest form.
Sometimes, the GCD will be greater than 1, and in those cases, dividing both the numerator and the denominator by the GCD will reduce the fraction further.

numerator and denominator

Every fraction is made up of two parts: the numerator and the denominator.
Understanding their roles is key to reducing fractions effectively.

• The numerator is the top number in a fraction. It represents the part of the whole you have.
• The denominator is the bottom number in a fraction. It shows how many parts the whole is divided into.

In the fraction \(\frac{15}{16}\), 15 is the numerator, and 16 is the denominator.
When working to simplify a fraction, check if there's a common factor between them.
If found, you can divide by it to reduce the fraction. In our given example, since the GCD is 1, the structure of the fraction does not change by division.
Both parts need to be critically examined when simplifying.

lowest terms

A fraction is in its lowest terms when you cannot divide the numerator and denominator by any number other than 1.
This means there are no common divisors left, making the fraction as simple as it can be.

• After calculating the GCD of a fraction, divide the numerator and the denominator by this value.
• If the result cannot be simplified further, it means you have achieved the lowest terms.

For example, while simplifying \(\frac{15}{16}\), we determined that its GCD is 1.
This signifies the fraction is already in its simplest form because dividing by 1 does not change its value.
Simplifying to the lowest terms ensures clarity and accuracy when working with fractions in math problems.