Problem 25 - Prealgebra

Question

Reduce each fraction to lowest terms. $$\frac{42}{66}$$

Step-by-Step Solution

Answer

The reduced fraction is \( \frac{7}{11} \).

Step 1: Identify the Greatest Common Divisor (GCD)

First, we need to find the greatest common divisor (GCD) of the numerator and denominator. Here, the numbers are 42 and 66. We can find the GCD by listing the factors of each number. The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. The factors of 66 are 1, 2, 3, 6, 11, 22, 33, and 66. The largest number common to both lists is 6, so the GCD is 6.

Step 2: Divide by the GCD

To reduce the fraction \( \frac{42}{66} \) to its lowest terms, divide both the numerator and the denominator by their GCD, which is 6. Performing this division gives: \( \frac{42 \div 6}{66 \div 6} = \frac{7}{11} \).

Step 3: Verify the Reduction

After dividing, we have the fraction \( \frac{7}{11} \). To verify that this fraction is in its lowest terms, check if there are any common factors other than 1 between 7 and 11. Since 7 and 11 are both prime numbers, they have no common factors other than 1, confirming the fraction is fully simplified.

Key Concepts

Greatest Common Divisor (GCD)

When simplifying fractions, finding the greatest common divisor (GCD) is a crucial step. The GCD is the largest positive integer that divides two numbers without leaving a remainder. For instance, in our fraction \( \frac{42}{66} \), we need to find the GCD of 42 and 66.

• First, list the factors of each number.
• The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42.
• The factors of 66 are: 1, 2, 3, 6, 11, 22, 33, 66.

The largest number that appears in both lists is the GCD. In this case, it is 6. With the GCD identified, you can proceed to reduce the fraction to its simplest form.

Prime Numbers

Prime numbers play a significant role when working with fractions, especially during simplification. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Understanding primes is key when breaking down numbers to find the GCD or checking if a fraction is in its simplest form. For example, after simplifying \( \frac{42}{66} \) to \( \frac{7}{11} \), we note that 7 and 11 are both prime numbers.

• This means that they each have only two divisors: 1 and the number itself.
• Thus, they share no common factors other than 1, ensuring that \( \frac{7}{11} \) is in its lowest terms.

Recognizing prime numbers can ease the task of simplifying fractions and verifying complete simplification.

Lowest Terms

A fraction is in its lowest terms when the numerator and the denominator have no common factors other than 1. Reducing a fraction to its lowest terms makes it simpler and easier to understand at a glance.
To bring a fraction to its lowest terms, divide both its numerator and denominator by their GCD. In our example, once we determined the GCD was 6, we divided:\( \frac{42}{66} = \frac{42 \div 6}{66 \div 6} = \frac{7}{11} \).

After reduction, ensure that there are no shared factors between the numerator and denominator. In \( \frac{7}{11} \), since both 7 and 11 are prime, the fraction is confirmed to be fully simplified. Always double-check to confirm you have reached the minimal form possible for accuracy in calculations and communication.