Problem 43 - Prealgebra

Question

Reduce each fraction to lowest terms. $$\frac{126}{165}$$

Step-by-Step Solution

Answer

The reduced fraction is \(\frac{42}{55}\).

Step 1: Find the Greatest Common Divisor (GCD)

Identify the greatest number that divides both the numerator (126) and the denominator (165) without leaving any remainder. - First, list the factors of 126, which are 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126. - Next, list the factors of 165, which are 1, 3, 5, 11, 15, 33, 55, 165. The common factors of both numbers are 1, 3. Therefore, the greatest common divisor (GCD) is 3.

Step 2: Divide Both Numerator and Denominator by the GCD

Using the GCD found in Step 1, divide both the numerator (126) and the denominator (165) by 3.- Divide the numerator: \( 126 \div 3 = 42 \)- Divide the denominator: \( 165 \div 3 = 55 \)The fraction \(\frac{126}{165}\) simplified by dividing by the GCD of 3 is \(\frac{42}{55}\).

Step 3: Confirm that the Fraction is in Lowest Terms

Verify that the new fraction \(\frac{42}{55}\) is in its lowest terms by checking that 42 and 55 have no common factors other than 1.- List factors of 42: 1, 2, 3, 6, 7, 14, 21, 42.- List factors of 55: 1, 5, 11, 55.Since the only common factor is 1, \(\frac{42}{55}\) is in its simplest form.

Key Concepts

Greatest Common Divisor (GCD)

The concept of the Greatest Common Divisor (GCD) may sound complex, but it's quite straightforward with a little practice. The GCD of two numbers is the largest number that can evenly divide both numbers without leaving a remainder. In simpler terms, it's the biggest number that both numbers share as a factor. This is a useful tool when you want to simplify fractions.

To find the GCD:

• List all factors of each number.
• Identify the common factors shared by both numbers.
• Select the largest of these shared numbers. This is the GCD.

For example, in the fraction \(\frac{126}{165}\), the GCD is 3. This means 3 is the largest number that divides both 126 and 165 evenly.

Factors of Numbers

Understanding factors is essential in simplifying fractions and calculating the GCD. A factor is a number that divides another number without causing any leftovers. Both multiplication and division are involved when dealing with factors.

For instance:

• The factors of 126 are 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, and 126.
• The factors of 165 are 1, 3, 5, 11, 15, 33, 55, and 165.

When two numbers are involved, find all factors for each number, then check which ones they have in common. If you're looking for the GCD, select the highest common value. This technique is not only helpful for simplifying fractions but also lays the groundwork for more advanced math concepts.

Simplifying Fractions

Simplifying fractions means adjusting the fraction so that the numerator and the denominator have no common factors other than 1. It makes the fraction easier to understand and work with. By doing this, you convert a fraction into its simplest form, making calculations cleaner and more manageable.

To simplify a fraction like \(\frac{126}{165}\):

• First, find the GCD. As earlier detailed, the GCD here is 3.
• Next, divide both the numerator and the denominator by that GCD.
• So, \(126 \div 3 = 42\) and \(165 \div 3 = 55\), simplifying the fraction to \(\frac{42}{55}\) .

After simplifying, double-check: list the factors of both resulting numbers to confirm they have no common factors other than 1. If none exist, your fraction is fully simplified. This practice not only hones your math skills but also ensures clarity in your calculations.