Problem 92 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{15}{1}$$

Step-by-Step Solution

Answer

The fraction \( \frac{15}{1} \) is already in its simplest form: \( 15 \).

Step 1: Identify the numerator and the denominator

In the fraction \( \frac{15}{1} \), the numerator is 15, and the denominator is 1.

Step 2: Find the greatest common divisor (GCD)

Since the denominator is 1, any number divided by 1 itself is the number. Therefore, the GCD of 15 and 1 is 1.

Step 3: Reduce the fraction

To reduce the fraction \( \frac{15}{1} \) to its lowest terms, divide both the numerator and the denominator by their GCD, which is 1. Thus, the reduced form is \( \frac{15}{1} \).

Step 4: Verify simplification

Check if any further simplification is possible. Since the GCD of the numerator and denominator is 1, the fraction is already in its simplest form.

Key Concepts

Numerator

In mathematics, a fraction is made up of two main parts. One of these parts is called the **numerator**. The numerator is the number located at the top of the fraction. It represents how many parts of the whole are being considered. For example, in the fraction \( \frac{15}{1} \), the number 15 is the numerator. This means we have 15 parts we are considering in this fraction.

When simplifying fractions, sometimes the numerator can be divided by the same number that divides the denominator. This process allows us to find an equivalent fraction that has smaller numbers but the same value. However, when the numerator is a specific number such as 15 and the denominator is 1, the fraction does not require further reduction since it already represents the whole value of 15.

Denominator

A fraction's **denominator** sits below the line of the fraction. It tells us into how many equal parts the whole is divided. So, in our example fraction \( \frac{15}{1} \), the number 1 is the denominator. This tells us that the whole is not divided into smaller parts but is taken as a complete unit.

A denominator of 1 means the fraction is already as simplified as it can get, because it shows that the fraction is simply counting the numerator. With any fraction with a denominator of 1, such as \( \frac{n}{1} \), where \( n \) is any number, the fraction itself equals \( n \). Therefore, it is essentially not a fraction in practical terms but rather a whole number.

Greatest Common Divisor (GCD)

The **greatest common divisor (GCD)** is a key concept in simplifying fractions. It is the largest number that can exactly divide both the numerator and the denominator without leaving a remainder. Finding the GCD is crucial when reducing fractions to their simplest form.

In the fraction \( \frac{15}{1} \), the process becomes straightforward because the denominator is 1. The GCD of any number and 1 is always 1 because 1 is a universal divisor. This means the fraction \( \frac{15}{1} \) can only be 'simplified' to \( \frac{15}{1} \) itself. When the GCD is 1, it indicates the fraction is already in its simplest form.

• Identifying the GCD involves checking common divisors of the numerators and denominators.
• Fractions like \( \frac{15}{1} \) inherently have 1 as the GCD due to the denominator being 1.
• Simplifying based on the GCD helps us ascertain if a fraction can be reduced further or is already reduced.