Problem 82 - Beginning and Intermediate Algebra

Question

Perform the following operations. Write answers in lowest terms. $$ \frac{8}{11}-\frac{1}{4}+\frac{1}{2} $$

Step-by-Step Solution

Answer

The result is \(\frac{43}{44}\).

Step 1: Identify the Least Common Denominator (LCD)

To perform operations on these fractions, we must have a common denominator. The denominators are 11, 4, and 2. The least common multiple of these numbers is 44, which will be our least common denominator (LCD).

Step 2: Convert Fractions to Have the Same Denominator

For each fraction, we find an equivalent fraction with the denominator 44. - Multiply \(\frac{8}{11}\) by \(\frac{4}{4}\) to get \(\frac{32}{44}\).- Multiply \(\frac{1}{4}\) by \(\frac{11}{11}\) to get \(\frac{11}{44}\).- Multiply \(\frac{1}{2}\) by \(\frac{22}{22}\) to get \(\frac{22}{44}\).

Step 3: Combine the Fractions

Add and subtract the numerators of the fractions over the common denominator 44. The operation is \(\frac{32}{44} - \frac{11}{44} + \frac{22}{44}\). First, subtract \(\frac{11}{44}\) from \(\frac{32}{44}\) giving \(\frac{21}{44}\), then add \(\frac{22}{44}\) resulting in \(\frac{43}{44}\).

Step 4: Simplify if Necessary

The fraction \(\frac{43}{44}\) is already in its lowest terms as 43 and 44 have no common factors other than 1.

Key Concepts

Least Common Denominator

When performing addition or subtraction with fractions, it's essential to have a common denominator. This concept is crucial because it allows comparison or combination of the fractions. The least common denominator (LCD) is the smallest number that can be a common denominator for two or more fractions. In our exercise, we have three denominators: 11, 4, and 2.
To find the LCD, you need to calculate the least common multiple (LCM) of these numbers. Here's a simple approach:

• List the multiples of each number until you find a common one.
• The LCM of 11, 4, and 2 is 44, as it's the smallest number divisible by all three.

By using 44 as the common denominator, you ensure that each fraction can be easily compared and calculated.

Subtracting Fractions

When subtracting fractions, the denominators must be the same. In our example, after converting each fraction to have the denominator of 44, we have: \(\frac{32}{44}\), \(\frac{11}{44}\), and \(\frac{22}{44}\). Subtraction is straightforward once the fractions have a common denominator.
To subtract \(\frac{11}{44}\) from \(\frac{32}{44}\):

• Keep the denominator (44).
• Subtract the numerators: 32 - 11 = 21.
• The result is \(\frac{21}{44}\).

Subtraction in fraction operations is no different from subtraction with whole numbers as long as the denominators match.

Adding Fractions

Adding fractions is similar to subtraction in terms of needing a common denominator first. Once we have the fractions with a common denominator, the process becomes simple.
In the example, after subtracting \(\frac{11}{44}\) from \(\frac{32}{44}\) and arriving at \(\frac{21}{44}\), you need to add \(\frac{22}{44}\).

• Keep the common denominator (44).
• Add the numerators: 21 + 22 = 43.
• The result is \(\frac{43}{44}\).

To ensure the final result is in its simplest form, check if the numerator and denominator have any common factors other than 1. Here, \(\frac{43}{44}\) is already in its lowest terms.