Problem 88 - Elementary Algebra

Question

Reduce each of the following to lowest terms. Don't forget that we reviewed some divisibility rules in Problem Set 1.2. (a) \(\frac{99}{117}\) (b) \(\frac{175}{225}\) (c) \(\frac{-111}{123}\) (d) \(\frac{-234}{270}\) (e) \(\frac{270}{495}\) (f) \(\frac{324}{459}\) (g) \(\frac{91}{143}\) (h) \(\frac{187}{221}\)

Step-by-Step Solution

Answer

The reduced fractions are \(\frac{11}{13}\), \(\frac{7}{9}\), \(\frac{-37}{41}\), \(\frac{-13}{15}\), \(\frac{6}{11}\), \(\frac{12}{17}\), \(\frac{7}{11}\), \(\frac{11}{13}\).

Step 1: Reduce \(\frac{99}{117}\)

Find the greatest common divisor (GCD) of 99 and 117, which is 9. Divide both the numerator and the denominator by 9. \[ \frac{99 \div 9}{117 \div 9} = \frac{11}{13} \].

Step 2: Reduce \(\frac{175}{225}\)

Find the GCD of 175 and 225, which is 25. Divide both the numerator and the denominator by 25. \[ \frac{175 \div 25}{225 \div 25} = \frac{7}{9} \].

Step 3: Reduce \(\frac{-111}{123}\)

First, ignore the negative sign and find the GCD of 111 and 123, which is 3. Keep track of the negative sign. Divide both the numerator and the denominator by 3. \[ \frac{-111 \div 3}{123 \div 3} = \frac{-37}{41} \].

Step 4: Reduce \(\frac{-234}{270}\)

Find the GCD of 234 and 270, which is 18. Ignore the negative sign for finding the GCD but keep it in your reduced fraction. Divide both the numerator and the denominator by 18. \[ \frac{-234 \div 18}{270 \div 18} = \frac{-13}{15} \].

Step 5: Reduce \(\frac{270}{495}\)

Find the GCD of 270 and 495, which is 45. Divide both the numerator and the denominator by 45. \[ \frac{270 \div 45}{495 \div 45} = \frac{6}{11} \].

Step 6: Reduce \(\frac{324}{459}\)

Find the GCD of 324 and 459, which is 27. Divide both the numerator and the denominator by 27. \[ \frac{324 \div 27}{459 \div 27} = \frac{12}{17} \].

Step 7: Reduce \(\frac{91}{143}\)

Find the GCD of 91 and 143, which is 13. Divide both the numerator and the denominator by 13. \[ \frac{91 \div 13}{143 \div 13} = \frac{7}{11} \].

Step 8: Reduce \(\frac{187}{221}\)

Find the GCD of 187 and 221, which is 17. Divide both the numerator and the denominator by 17. \[ \frac{187 \div 17}{221 \div 17} = \frac{11}{13} \].

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is a crucial component when working with fractions, particularly when reducing them to their simplest form. The GCD of two numbers is the largest number that can divide both of them without leaving a remainder. To find the GCD:

• List the factors of each number.
• Identify the common factors shared by both numbers.
• Select the greatest of these common factors.

To understand it better, let's take the example from our exercise: for the numbers 99 and 117, their GCD is 9 because 9 is the largest number that divides evenly into both. This concept is important because dividing the numerator and the denominator of a fraction by their GCD reduces the fraction to its simplest form while retaining equivalency.

Fraction Simplification

Simplifying fractions is a key process in making fractions easier to understand and work with. A fraction is simplified when both the numerator and the denominator are divided by their greatest common divisor (GCD), yielding the simplest form of the fraction. Here's a step-by-step approach:

• Identify the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• Ensure that the resulting fraction cannot be simplified further.

For instance, the exercise shows that \(\frac{99}{117}\) can be simplified to \(\frac{11}{13}\) by dividing the numerator and denominator by their GCD, which is 9. Simplifying a fraction ensures that it is as condensed as possible, while maintaining its proportionate value. This becomes useful in comparing fractions, performing operations like addition or subtraction, and solving equations.

Divisibility Rules

Divisibility rules are helpful shortcuts that simplify the process of finding common divisors, especially when attempting to simplify fractions. They provide easy ways to determine if a number is divisible by another without performing long division. Here are some basic rules that can aid in finding the GCD:

• A number is divisible by 2 if its last digit is even.
• A number is divisible by 3 if the sum of its digits is divisible by 3.
• A number is divisible by 5 if its last digit is 0 or 5.
• A number is divisible by 9 if the sum of its digits is divisible by 9.

Using these rules can significantly reduce the time it takes to find factors and, consequently, the greatest common divisor. For example, in our exercise with \(\frac{175}{225}\), applying the divisibility rule for 5 instantly reveals common factors, since both numbers end in 5. Understanding these rules streamlines the process of fraction simplification and helps clarify how numbers can interact without relying solely on intensive calculations.