Problem 22 - Elementary Algebra

Question

For the following problems, reduce, if possible, each fraction lowest terms. \(\frac{44}{11}\)

Step-by-Step Solution

Answer

Question: Reduce the fraction \(\frac{44}{11}\) to its lowest terms. Answer: The simplified fraction is \(\frac{44}{11} = \frac{4}{1} = 4\), which is a whole number.

Step 1: Find the Greatest Common Divisor (GCD) of 44 and 11

To find the GCD of 44 and 11, list the factors of each number and identify the largest factor that both numbers have in common. The factors of 44 are 1, 2, 4, 11, 22, and 44. The factors of 11 are 1 and 11. The greatest common divisor of 44 and 11 is 11.

Step 2: Divide both the numerator and denominator by the GCD

Now that we have identified the GCD as 11, we will divide both the numerator and denominator of the fraction \(\frac{44}{11}\) by 11. \(\frac{44}{11} = \frac{44 ÷ 11}{11 ÷ 11} = \frac{4}{1}\)

Step 3: Write down the simplified fraction

The simplified fraction is \(\frac{4}{1}\). Since the denominator is 1, the fraction represents the whole number 4. Therefore, the reduced fraction, in this case, is the whole number 4.

Key Concepts

Understanding the Greatest Common Divisor

The Greatest Common Divisor (GCD) is a key concept when it comes to simplifying fractions. It is the largest number that divides two or more numbers without leaving a remainder. Finding the GCD is a vital first step in the process of fraction simplification. For example, when simplifying the fraction \( \frac{44}{11} \), identifying the GCD of 44 and 11 helps us determine the largest factor these two numbers share which is 11.

Here's how you can find the GCD:

• List the factors of each number.
• Identify the common factors.
• Select the largest common factor.

This method ensures that the fraction is reduced to its simplest form. By dividing both the numerator and the denominator by the GCD, you achieve the simplified fraction. This process is essential for making fractions easier to understand and compare.

The Role of Numerators and Denominators

Numerators and denominators are the two main components of any fraction. Understanding their roles can help in the simplification process.

The numerator is the top part of the fraction and indicates how many parts we are considering. In the fraction \( \frac{44}{11} \), 44 is the numerator. The denominator, on the other hand, is the bottom part of the fraction and shows the total number of equal parts into which something is divided. Here, 11 is the denominator.

To simplify a fraction, we aim to make these numbers as small as possible while still keeping their ratio the same. By dividing both the numerator and the denominator by their GCD, you can convert the fraction into its simplest form, which makes it more intuitive to work with or interpret. This action brings clarity and simplicity to mathematical operations.

From Simplified Fractions to Whole Numbers

When a fraction simplifies to a form where the denominator becomes 1, it becomes a whole number. This is what happens in the case of \( \frac{44}{11} \), which simplifies to \( \frac{4}{1} \). Since dividing any number by 1 is the number itself, \( \frac{4}{1} \) equals 4, a whole number.

Whole numbers are numbers without fractions or decimals. They are part of our everyday counting numbers and include zero and positive integers. Understanding when a fraction equates to a whole number is crucial in mathematics as it relates to practical scenarios where counts of items are exact and without parts. Simplifying to whole numbers can make workings and answers clearer, particularly when dealing with measurements or quantifiable data.