Problem 90 - Fundamentals of Mathematics

Question

For the following problems, reduce, if possible, each of the fractions to lowest terms. $$\frac{44}{11}$$

Step-by-Step Solution

Answer

The fraction \( \frac{44}{11} \) simplifies to 4.

Step 1: Find the Greatest Common Divisor (GCD)

To reduce the fraction \( \frac{44}{11} \) to its lowest terms, start by finding the greatest common divisor (GCD) of the numerator and the denominator. Here, the GCD of 44 and 11 is 11.

Step 2: Divide by the GCD

Divide both the numerator and the denominator by their GCD, which is 11. Thus, \( \frac{44}{11} = \frac{44 \div 11}{11 \div 11} = \frac{4}{1} \).

Step 3: Simplify the Fraction

The fraction \( \frac{4}{1} \) can be further simplified to just 4, because any number divided by 1 is the number itself.

Key Concepts

Greatest Common Divisor

When working with fractions, the greatest common divisor, or GCD, is a crucial concept. It helps us simplify fractions by finding the largest number that divides both the numerator and the denominator. Think of it as the largest piece that both numbers share. For example, in the fraction \( \frac{44}{11} \), we need to find the GCD of 44 and 11.

• List all the divisors of 44: 1, 2, 4, 11, 22, 44.
• List all the divisors of 11: 1, 11.

The greatest number in both lists is 11. This means 11 is the GCD of 44 and 11. Knowing the GCD allows us to reduce the fraction to its simplest form by dividing both the numerator and the denominator by this number.

Numerator

In any fraction, the numerator is the top number. It's the number above the line in a fraction, and it represents how many parts of the whole are being considered. Let's take \( \frac{44}{11} \) again. Here, 44 is the numerator.
The numerator tells us

• How many parts we have.

When simplifying fractions, we divide the numerator by the GCD. So in our example, you divide the numerator 44 by 11 to get 4. This shows that after simplification, there are 4 whole parts remaining.

Denominator

The denominator is the bottom number in a fraction and tells us how many equal parts the whole is divided into. In \( \frac{44}{11} \), 11 is the denominator, meaning the whole is divided into 11 equal pieces.

• It shows the total number of parts.

When simplifying, after determining the GCD, you need to divide this denominator by the GCD. For \( \frac{44}{11} \), divide 11 by 11, which simplifies the denominator to 1. This is why \( \frac{4}{1} \) simplifies to 4.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. This is achieved by using the greatest common divisor (GCD).
Here's how you do it:

• Find the GCD of the numerator and denominator.
• Divide both by the GCD.

For example, in \( \frac{44}{11} \):

• The GCD is 11.
• Divide the numerator (44) and the denominator (11) by 11, resulting in \( \frac{4}{1} \).

Since any number over 1 is just the number itself, \( \frac{4}{1} \) simplifies further to 4. Thus, the simplest form of \( \frac{44}{11} \) is 4.