Problem 38 - Prealgebra

Question

Reduce each fraction to lowest terms. $$\frac{45}{75}$$

Step-by-Step Solution

Answer

The fraction \( \frac{45}{75} \) reduces to \( \frac{3}{5} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To reduce a fraction, we first find the greatest common divisor (GCD) of the numerator and the denominator. For the fraction \( \frac{45}{75} \), we need the GCD of 45 and 75.

Step 2: Calculate the GCD

List the factors of both numbers. The factors of 45 are: 1, 3, 5, 9, 15, 45. The factors of 75 are: 1, 3, 5, 15, 25, 75. The greatest common factor is 15.

Step 3: Divide Numerator and Denominator by the GCD

Divide both the numerator and the denominator by the GCD (15). This gives: \( \frac{45 \div 15}{75 \div 15} = \frac{3}{5} \).

Step 4: Verify the Result

Check that the fraction \( \frac{3}{5} \) is in its simplest form. The numerator 3 and denominator 5 have no common divisors other than 1, confirming it cannot be simplified further.

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) plays a crucial role in fraction simplification. It is the largest positive integer that divides both numbers, leaving no remainder. In other words, it's the biggest number that can exactly fit into both the numerator and the denominator. To identify the GCD:

• List the factors of each number.
• Find the largest factor common to both lists.

For example, in the fraction \( \frac{45}{75} \), we listed the factors of 45 (1, 3, 5, 9, 15, 45) and the factors of 75 (1, 3, 5, 15, 25, 75). The largest number that appears in both lists is 15, so the GCD is 15.

Numerator

The numerator is the top number in a fraction. It tells us how many parts of a whole we are considering. In the fraction \( \frac{45}{75} \), the numerator is 45. When simplifying a fraction, we divide the numerator by the GCD. Continuing with our example, we divide 45 by the GCD, 15, giving us 3. This is a key part of reducing fractions to their simplest form, helping to make the values clearer and easier to work with.

Denominator

The denominator is the bottom number of a fraction. It indicates the total number of equal parts into which the whole is divided. In the fraction \( \frac{45}{75} \), the denominator is 75. During simplification, we must also divide the denominator by the GCD. For this example, dividing 75 by the GCD, which is 15, results in 5. With both the numerator and denominator divided by the GCD, the fraction becomes simplified, showing the same value but in an easier-to-manage form.

Simplest Form

A fraction is in its simplest form when the numerator and denominator have no common divisors other than 1. This means that the fraction cannot be simplified further. To verify if a fraction is in its simplest form:

• Check that the GCD of the numerator and the denominator is 1.
• If it is, the fraction is as simple as it gets.

In the reduced form of \( \frac{45}{75} \), the simplified fraction \( \frac{3}{5} \) has a numerator and a denominator with no common factors besides 1. Therefore, \( \frac{3}{5} \) is indeed the simplest form of the original fraction.