Problem 52 - Fundamentals of Mathematics

Question

Reduce, if possible, each fraction. $$\frac{18}{25}$$

Step-by-Step Solution

Answer

The fraction \( \frac{18}{25} \) is already in its simplest form.

Step 1: Identify the Numerator and Denominator

The given fraction is \( \frac{18}{25} \). Here, the numerator is 18 and the denominator is 25.

Step 2: Find the Greatest Common Divisor (GCD)

Determine the greatest common divisor for the numbers 18 and 25. The GCD is the largest number that divides both numbers without leaving a remainder.

Step 3: Check for Common Factors

List the factors of 18: 1, 2, 3, 6, 9, 18. List the factors of 25: 1, 5, 25. The only common factor is 1.

Step 4: Determine Reducibility

Since the greatest common divisor is 1, \( \frac{18}{25} \) has no other common factors besides 1. This means the fraction cannot be reduced further.

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is a crucial concept in simplifying fractions. To find the GCD, we look for the largest number that can evenly divide both the numerator and the denominator. This means that the GCD is the biggest number that leaves no remainder when it divides both numbers.

For example, consider two numbers, 18 and 25. In this instance, the process would go like this:

• List all the factors of 18: 1, 2, 3, 6, 9, 18.
• List all the factors of 25: 1, 5, 25.

Here, you can see that the largest number that appears in both lists is 1. Hence, the GCD of 18 and 25 is 1.

When the GCD is 1, it tells us the fraction cannot be reduced further, as the numerator and the denominator have no larger common factor.

Numerator and Denominator

In a fraction, the numerator and denominator are the two key components. The numerator is the top number, representing how many parts we have. The denominator is the bottom number, indicating the total number of equal parts in a whole.

In our fraction example, \( \frac{18}{25} \):

• The numerator is 18, showing the number of parts considered.
• The denominator is 25, indicating the total parts that make up the whole.

Understanding their roles helps us evaluate and simplify fractions better. Always look at these two numbers when considering how to simplify a fraction, checking for common factors or using the greatest common divisor.

Common Factors

Common factors are essential in the process of simplifying fractions. A common factor is any number that can divide two numbers without leaving a remainder. Finding these numbers helps us understand how the numbers in a fraction relate to each other.

For instances like \( \frac{18}{25} \), identifying common factors works as follows:

• List the factors of 18: 1, 2, 3, 6, 9, 18.
• List the factors of 25: 1, 5, 25.

Examining these lists shows that 1 is the only factor shared by both 18 and 25. This means 18 and 25 have no other larger common factors, implying the fraction is already in its simplest form.

When two numbers share no factors other than 1, they are coprime. Hence, the fraction cannot be reduced further beyond its current state.