Problem 49 - Fundamentals of Mathematics

Question

Reduce, if possible, each fraction. $$\frac{32}{44}$$

Step-by-Step Solution

Answer

\(\frac{32}{44}\) reduces to \(\frac{8}{11}\).

Step 1: Identify the Greatest Common Divisor (GCD)

First, we need to find the greatest common divisor (GCD) of the numerator and the denominator, which are 32 and 44. To do this, factor both numbers into their prime factors:- 32 factors into: \(2^5\)- 44 factors into: \(2^2 \times 11\)The common factor is \(2^2\), which is 4. Therefore, the GCD of 32 and 44 is 4.

Step 2: Divide Numerator and Denominator by the GCD

Now that we have the GCD, we divide both the numerator and the denominator by it to simplify the fraction:\(\frac{32}{44} = \frac{32 \div 4}{44 \div 4} = \frac{8}{11}\).

Step 3: Check If the Fraction is Fully Simplified

The resulting fraction is \(\frac{8}{11}\). We check if it can be simplified further by ensuring the numerator and denominator have no common factors other than 1. Because 8 and 11 have no common factors (8 is even and 11 is prime), \(\frac{8}{11}\) is fully simplified.

Key Concepts

Understanding the Greatest Common Divisor (GCD)

To simplify fractions, finding the greatest common divisor (GCD) of the numerator and denominator is crucial. The GCD is the largest number that divides both numbers without leaving a remainder.

Here's why the GCD matters:

• It helps us simplify fractions to their lowest terms, making them easier to understand and work with.
• Identifying the GCD involves a well-organized process, such as prime factorization, to ensure accuracy.

In the fraction \( \frac{32}{44} \), we used the GCD of 4 to reduce it. By understanding the GCD's role, we ensure that the fraction is as simple as possible.

Exploring Prime Factorization

Prime factorization involves breaking down a number into its basic prime factors. Prime numbers are numbers that have only two divisors: 1 and themselves, such as 2, 3, 5, etc. This method helps in finding the GCD efficiently.

Here’s how we can use prime factorization:

• For 32: we split 32 into its prime factors which are \(2^5 \), as it’s made up of five 2s multiplied together.
• For 44: we decompose it into \(2^2 \times 11\), indicating it's made of two 2s and one 11.

To find the GCD, we take the lowest power of all common primes. Here, both 32 and 44 share the prime 2, and the smallest power is \(2^2\), which equals 4. This shows prime factorization’s power in simplifying processes.

Numerator and Denominator: The Building Blocks of Fractions

Fractions are expressions that represent parts of a whole. They have two key components: the numerator and the denominator.

Understanding these components helps in simplifying fractions:

• The numerator (32 in our example) is the top number reflecting how many parts we have.
• The denominator (44 in our example) is the bottom number showing into how many equal parts the whole is divided.

By dividing both the numerator and denominator by their GCD, we find a simpler form. In our problem, dividing 32 and 44 by their GCD, which is 4, gives us \( \frac{8}{11} \). This form is more readable and easier to work with, reflecting a true, simplified relationship between the parts and the whole without altering the fraction's value.