Problem 44 - Fundamentals of Mathematics

Question

Determine if the pairs of fractions are equivalent. $$\frac{8}{9}, \frac{32}{36}$$

Step-by-Step Solution

Answer

The fractions are equivalent.

Step 1: Express Both Fractions

Express the two given fractions. The fractions given are \( \frac{8}{9} \) and \( \frac{32}{36} \). Our goal is to determine if these fractions are equivalent.

Step 2: Simplify the Second Fraction

To check if the fractions are equivalent, simplify \( \frac{32}{36} \) to its simplest form by finding the greatest common divisor (GCD) of 32 and 36, which is 4. Divide the numerator and the denominator by their GCD: \( \frac{32 \div 4}{36 \div 4} = \frac{8}{9} \).

Step 3: Compare Simplified Fractions

After simplifying, \( \frac{32}{36} = \frac{8}{9} \). Now, compare the simplified form of \( \frac{32}{36} \) with \( \frac{8}{9} \). Since they are the same, the fractions are equivalent.

Key Concepts

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator (top number) and the denominator (bottom number) have no common divisors other than 1. This makes the fraction easier to work with and understand.

• To simplify a fraction, find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.
• The result is a fraction in its simplest form.

For instance, let's simplify the fraction \( \frac{32}{36} \). By finding the GCD of 32 and 36, we get 4. Then we divide both 32 and 36 by 4, giving us \( \frac{8}{9} \). Thus, the simplified version of \( \frac{32}{36} \) is \( \frac{8}{9} \). Always verify the simplified fraction to ensure there are no common divisors left apart from 1.

Greatest Common Divisor

The greatest common divisor (GCD), sometimes called the greatest common factor, is the largest positive integer that divides two or more numbers without leaving a remainder. Finding the GCD is crucial when simplifying fractions because it tells us what number we can use to reduce both the numerator and the denominator.
Here’s how to find the GCD:

• List the factors of each number.
• Identify the common factors between these lists.
• The greatest of these common factors is the GCD.

In our example, to find the GCD of 32 and 36, we list their factors:
- Factors of 32 are 1, 2, 4, 8, 16, 32.- Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.
The common factors are 1, 2, and 4. The greatest of these is 4, so the GCD is 4. This GCD lets us simplify the fraction \( \frac{32}{36} \) to \( \frac{8}{9} \). Using the GCD ensures the fraction is reduced to its simplest form.

Comparing Fractions

Comparing fractions involves determining if different fractions are equivalent or which one is larger or smaller. To compare fractions directly, it is most useful to have them in their simplest form.
When fractions are in their simplest form:

• If the numerators and denominators are equal, the fractions are equivalent.
• If the denominators are the same, compare numerators directly.
• If the numerators are the same, compare denominators directly (a smaller denominator means a larger fraction).
• If neither numerators nor denominators match, you may need to find a common denominator or cross-multiply to compare them.

In the given exercise, we simplified \( \frac{32}{36} \) to \( \frac{8}{9} \). Then we compared it to the original fraction \( \frac{8}{9} \). Since these fractions are identical when simplified, they are equivalent. Thus, through simplification, we confirm \( \frac{8}{9} = \frac{8}{9} \), endorsing their equivalence. Using simplification and comparison helps verify relationships between fractions accurately.