Problem 50 - Elementary and Intermediate Algebra

Question

Simplify. $$ \frac{75}{45} $$

Step-by-Step Solution

Answer

\( \frac{5}{3} \)

Step 1: Identify the Greatest Common Divisor (GCD)

Find the largest number that divides both the numerator (75) and the denominator (45) without leaving a remainder. The GCD of 75 and 45 can be found by factorizing each number: 75 = 3 * 5^2 and 45 = 3^2 * 5. The common factors are 3 and 5.

Step 2: Divide Numerator and Denominator by the GCD

The greatest common divisor of 75 and 45 is 15. Divide the numerator and the denominator by this GCD. \[\frac{75 ÷ 15}{45 ÷ 15} = \frac{5}{3}\]

Step 3: Simplified Fraction

The fraction simplified by dividing both the numerator and the denominator by their greatest common divisor (15) is \( \frac{5}{3} \). This process gives us the final simplified form.

Key Concepts

Greatest Common Divisor

The Greatest Common Divisor (GCD) is essential when simplifying fractions. It is the largest number that can evenly divide both the numerator and the denominator of a fraction. Without the GCD, we wouldn't know the most accurate way to simplify a fraction.
To find the GCD of two numbers, follow these steps:

• Factorize both numbers into their prime factors.
• Identify the common factors.
• Multiply the common factors to find the GCD.

For example, in the problem \( \frac{75}{45} \), we factorized 75 into \(3 \times 5^2\) and 45 into \(3^2 \times 5\). The common factors are 3 and 5, giving us a GCD of 15. Dividing both the numerator and the denominator by this GCD simplifies the fraction.

Numerator and Denominator

In a fraction, the numerator is the number on the top, and the denominator is the number on the bottom. The numerator represents how many parts we have, while the denominator shows how many parts make a whole.
For example, in \( \frac{75}{45} \), 75 is the numerator and 45 is the denominator. Understanding these positions helps us apply the correct steps in fraction simplification.
When we simplify a fraction, we divide both the numerator and the denominator by the same number to reduce it to its simplest form. Ensure you always divide both terms by their GCD to maintain the fraction's value.

Factorization

Factorization is breaking down a number into its prime factors. These are numbers that can only be divided by 1 and themselves. Factorization is crucial in finding the GCD for simplifying fractions.
For instance, to factorize 75, we find the numbers that multiply to give 75: 3 and \(5^2\). Similarly, 45 factorizes into \(3^2\) and 5. Identifying common factors, such as 3 and 5 in this example, allows us to determine the GCD.
Once we've factorized both the numerator and the denominator, we proceed to simplify the fraction by dividing both by their GCD.