Problem 45 - Elementary Algebra

Question

For exercises 39-46, rewrite the percent as a fraction. Simplify the fraction into lowest terms. $$ 113 \% $$

Step-by-Step Solution

Answer

\(\frac{113}{100}\)

Step 1: Understand the problem

The task is to convert a percentage to a fraction and then simplify the fraction. In this case, the percentage given is 113%.

Step 2: Convert percent to a fraction

To convert 113% into a fraction, write it over 100 as follows: 113% = \(\frac{113}{100}\).

Step 3: Simplify the fraction

Check if the fraction \(\frac{113}{100}\) can be simplified further. To do this, find the greatest common divisor (GCD) of 113 and 100. Since 113 is a prime number and does not have common divisors with 100 other than 1, the fraction is already in its simplest form.

Key Concepts

Fraction Simplification

Converting a percentage to a fraction and then simplifying it is a common math exercise. The process involves expressing the percentage over 100 and then reducing the resulting fraction to its simplest form. Simplifying a fraction means reducing it to the smallest possible numerator and denominator without changing the value of the fraction. To do this, you need to find the Greatest Common Divisor (GCD) of the numerator and denominator and divide both by this number. For example, if you start with the percentage 113%, you write it as \(\frac{113}{100}\). Check if this fraction can be simplified further by identifying the GCD of 113 and 100, but, as we will see in subsequent sections, 113 being a prime means the fraction is already in its lowest terms.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest number that divides both the numerator and the denominator of a fraction without leaving a remainder. Finding the GCD is crucial for simplifying fractions. To find the GCD of two numbers, you can use several methods, including:

• Listing out the factors
• Using the Euclidean algorithm

For instance, to find the GCD of 113 and 100, list the factors of each:

• Factors of 113: 1, 113
• Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100

Here, 1 is the only common factor, making the GCD 1. This indicates that the fraction \(\frac{113}{100}\) is already simplified as much as possible.

Prime Numbers

Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. In other words, a prime number has exactly two distinct positive divisors: 1 and the number itself. Learning about prime numbers is essential in understanding why some fractions can't be simplified further.
For example, 113 is a prime number because it only has two divisors: 1 and 113. Because its only divisors are 1 and itself, 113 cancels no common factors with 100 (other than 1). As a result, the fraction \(\frac{113}{100}\) cannot be reduced further, meaning the fraction is already in its simplest form. Recognizing prime numbers helps you quickly determine when to stop trying to simplify a fraction.