Problem 96 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{45}{32}$$

Step-by-Step Solution

Answer

The fraction \( \frac{45}{32} \) is already in its simplest form.

Step 1: Identify the GCD

To reduce the fraction \( \frac{45}{32} \) to its lowest terms, we first need to find the greatest common divisor (GCD) of the numerator (45) and the denominator (32). Use the method of finding the highest factor that both numbers share.

Step 2: Check Factors of 45 and 32

The factors of 45 are 1, 3, 5, 9, 15, and 45. The factors of 32 are 1, 2, 4, 8, 16, and 32. The greatest common factor between them is 1, as they have no other common factors.

Step 3: Simplify the Fraction

Since the GCD is 1, \( \frac{45}{32} \) is already in its simplest form. There are no factors other than 1 that divide both 45 and 32.

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is an essential concept in mathematics used to simplify fractions, among many other things. When trying to reduce a fraction like \( \frac{45}{32} \), the GCD plays a crucial role.The GCD is the largest positive integer that divides two or more numbers without leaving a remainder. To identify the GCD, you determine the "highest factor" that is common to both the numerator and the denominator. In simpler terms, you're looking for the largest number that can evenly go into both numbers. For instance:

• The GCD of 8 and 12 is 4, because 4 is the largest number that divides both 8 and 12 without a remainder.
• The GCD of 9 and 15 is 3, as 3 is the largest number that divides both.

To find the GCD, one can list out all factors of each number and compare which ones they share. Alternatively, advanced methods like the Euclidean algorithm are used, but for basic fractions, listing factors suffices.

Simplification of Fractions

Simplifying fractions is a fundamental skill in mathematics that allows you to express a fraction in its lowest terms. Essentially, you make a fraction as simple as possible by using the greatest common divisor to divide both the numerator and the denominator.For example, consider the fraction \( \frac{10}{15} \). To simplify it:1. Find the GCD, which is 5.2. Divide both the numerator and the denominator by 5.3. This provides the simplified fraction: \( \frac{2}{3} \).In the case of our fraction \( \frac{45}{32} \), the GCD is 1. This means there isn’t a larger number that can divide both 45 and 32 evenly, except for 1. Therefore, the fraction is already in its simplest form, and further simplification is unnecessary. Simplification helps in recognizing equal fractions, performing operations, and highlighting relationships between numbers.

Factors of Numbers

Factors are numbers we multiply together to get another number. They are the "building blocks" of numbers themselves. When you're dealing with numbers in a fraction, understanding their factors helps in the process of finding the GCD.To discover the factors of a number:

• Example: For 45, the factors are 1, 3, 5, 9, 15, and 45. These are all numbers that divide 45 without a remainder.
• Similarly, for 32, the factors are 1, 2, 4, 8, 16, and 32.

These numbers can only be divided by these factors without leaving a fraction or decimal. Often, identifying a common factor in two numbers is the first step to finding the GCD. In fractions, factors are useful for recognizing which numbers can simplify fractions, as they reveal divisibility. When no common factors exist other than 1, as in \( \frac{45}{32} \), the fraction is considered to be in its simplest form.