Problem 102 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{75}{45}$$

Step-by-Step Solution

Answer

The simplest form of \(\frac{75}{45}\) is \(\frac{5}{3}\).

Step 1: Finding the Greatest Common Divisor

To reduce the fraction \(\frac{75}{45}\) to its lowest terms, we first need to find the greatest common divisor (GCD) of the numerator 75 and the denominator 45. We list the factors:Factors of 75: 1, 3, 5, 15, 25, 75Factors of 45: 1, 3, 5, 9, 15, 45The greatest common factor they share is 15.

Step 2: Dividing Both Terms by the GCD

Now that we have identified 15 as the GCD of both 75 and 45, we divide both the numerator and the denominator of the fraction by 15:\[\frac{75 \div 15}{45 \div 15} = \frac{5}{3}\]

Step 3: Simplification and Verification

The fraction \(\frac{5}{3}\) is in its simplest form as there are no common divisors greater than 1 between 5 and 3. We can also verify by checking that the only divisor common to 5 and 3 is 1.

Key Concepts

Understanding the Greatest Common Divisor (GCD)

The greatest common divisor, often abbreviated as GCD, is a crucial concept when it comes to simplifying fractions. The GCD is the largest number that can exactly divide two or more numbers without leaving a remainder. To simplify a fraction like \(\frac{75}{45}\), we start by finding the GCD of the numerator and the denominator.

First, list all factors for both numbers:

• Factors of 75: 1, 3, 5, 15, 25, 75
• Factors of 45: 1, 3, 5, 9, 15, 45

The common factors are 1, 3, 5, and 15, but the greatest one is 15. Therefore, 15 is the GCD. So, understanding the GCD helps us know by how much we can reduce the fraction to its simplest form.

Numerator and Denominator Explained

In any fraction, you will always find two parts: the numerator and the denominator. The numerator is the top number, and it tells us how many parts we have. The denominator, the bottom number, tells us into how many equal parts the whole is divided.

For example, in \(\frac{75}{45}\), 75 is the numerator and 45 is the denominator. It means you have 75 parts out of 45 equal parts. To reduce this fraction, we need to reset the balance between the numerator and the denominator to a smaller, equivalent fraction. The GCD aids us in adjusting these numbers efficiently.

Simplifying Fractions to Their Lowest Terms

Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. To simplify \(\frac{75}{45}\), we've already determined the GCD, which is 15. We divide both parts of the fraction by this number.

Performing this division gives us:

• \(75 \div 15 = 5\)
• \(45 \div 15 = 3\)

So, \(\frac{75}{45}\) simplifies to \(\frac{5}{3}\). After simplification, always check that there are no larger common divisors. Here, 5 and 3 have no common divisors other than 1, confirming \(\frac{5}{3}\) is indeed the simplest form. Simplifying helps us work with more manageable numbers in equations and comparisons.