Problem 55 - Fundamentals of Mathematics

Question

Reduce, if possible, each fraction. $$\frac{45}{85}$$

Step-by-Step Solution

Answer

The fraction reduces to \( \frac{9}{17} \).

Step 1: Find the Greatest Common Divisor (GCD)

To reduce the fraction \( \frac{45}{85} \), we first need to find the greatest common divisor (GCD) of the numerator and the denominator. The factors of 45 are: 1, 3, 5, 9, 15, 45 and the factors of 85 are: 1, 5, 17, 85. The GCD is the largest common factor, which is 5.

Step 2: Divide Numerator and Denominator by the GCD

Now, divide both the numerator and the denominator by their GCD, which is 5. Calculate \( \frac{45}{85} = \frac{45 \div 5}{85 \div 5} = \frac{9}{17} \).

Step 3: Confirm the Result

Check the simplified fraction \( \frac{9}{17} \) to ensure there are no common divisors other than 1. The factors of 9 are: 1, 3, 9 and the factors of 17 are: 1, 17. There are no other common factors, confirming that \( \frac{9}{17} \) is in its simplest form.

Key Concepts

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is a crucial concept in understanding how to simplify fractions. When we look at two numbers, the GCD is the largest integer that can evenly divide both of them. Let's break it down:

• To find the GCD, list out all the factors of each number. Factors are numbers that you can multiply together to get the original number.
• Then, identify the largest factor that appears in both lists. This is your GCD.

By discovering the GCD, you unlock the ability to break down fractions into their simplest form. For example, in the fraction \( \frac{45}{85} \), the factors help determine that 5 is the GCD. Without the GCD, you would struggle to efficiently reduce the fraction.

numerator and denominator

In fractions, the terms "numerator" and "denominator" are key to understanding its structure. A fraction consists of two parts separated by a horizontal line:

• The number above the line is called the numerator. It represents how many parts of the whole we are considering.
• The number below the line is the denominator. It shows the total number of equal parts the whole is divided into.

Using the example \( \frac{45}{85} \), 45 is the numerator, and 85 is the denominator. When you're simplifying a fraction or finding the GCD, you'll always be working with both the numerator and the denominator.

simplifying fractions

Simplifying fractions is a technique of reducing a fraction to its simplest form. This means that the numerator and the denominator have no common divisors other than 1. Here's how you can simplify a fraction:

• First, find the Greatest Common Divisor (GCD) of the numerator and the denominator.
• Then, divide both the numerator and the denominator by the GCD.

In the case of \( \frac{45}{85} \), the GCD is 5. When we divide both 45 and 85 by 5, we end up with \( \frac{9}{17} \). Simplifying not only makes numbers easier to work with but also helps in comparing fractions. By ensuring there are no common factors left, \( \frac{9}{17} \) is the simplest form of the original fraction.