Problem 95 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Write the answer as a fraction or mixed number in simplest form. \(\frac{5}{11}-\frac{4}{5}+\frac{3}{4}\)

Step-by-Step Solution

Answer

The simplest form of the expression \(\frac{5}{11}-\frac{4}{5}+\frac{3}{4}\) is \(\frac{89}{220}\)

Step 1: Find the Least Common Denominator (LCD)

Locate the least common denominator, which is the smallest number that each of the three denominators (11,5 and 4) can divide into evenly. For this problem, the LCD is 220.

Step 2: Convert each fraction to an equivalent fraction with the LCD

For each fraction, divide the LCD by the fraction's denominator and then multiply both the numerator and denominator of the fraction by the result: \(\frac{5}{11}=\frac{5 \times 20}{11 \times 20}=\frac{100}{220}\), \(\frac{4}{5}=\frac{4 \times 44}{5 \times 44}=\frac{176}{220}\), and \(\frac{3}{4}=\frac{3 \times 55}{4 \times 55}=\frac{165}{220}\)

Step 3: Compute the operation

Perform the operation based on the new fractions: \(\frac{100}{220}-\frac{176}{220}+\frac{165}{220}=\frac{89}{220}\).

Step 4: Simplify the Result

The fraction \(\frac{89}{220}\) does not simplify as there is no number that divides evenly into both 89 and 220.

Key Concepts

Least Common Denominator

When adding or subtracting fractions, it's crucial to have a common denominator. This allows us to compare and combine the fractions easily. The least common denominator (LCD) is the smallest multiple that the denominators can divide into evenly. For instance, if you have fractions with denominators 11, 5, and 4, you need to find a number that all these numbers can go into. In this case, 220 is the least common denominator. Once you find the LCD, you can rewrite each fraction so they have the same denominator.

Steps to find the LCD:

• List multiples of each denominator.
• Identify the smallest multiple that all denominators share.
• Use this value to convert fractions to have this common denominator.

Finding the LCD aligns the denominators and sets the stage for adding or subtracting the fractions efficiently.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same value. After finding the least common denominator, you need to convert the given fractions into equivalent fractions with the LCD. By doing so, you're transforming the fractions without altering their value; you're merely changing their appearance.

Here's how you convert to equivalent fractions:

• Divide the LCD by the original denominator of each fraction.
• Multiply both the numerator and the denominator by this resulting number.

For example, to convert \(\frac{5}{11}\) into an equivalent fraction with a denominator of 220, you calculate \(\frac{5 \times 20}{11 \times 20} = \frac{100}{220}\). Do this for all original fractions to efficiently perform addition or subtraction.

Simplifying Fractions

Once you have performed the operation with the equivalent fractions, the next step is simplifying the final fraction. Simplifying a fraction means making it as simple as possible while maintaining its value. The simplest form of a fraction is when the numerator and the denominator have no common factors other than 1.

To simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD.

In the given problem, the final fraction is \(\frac{89}{220}\). Since 89 is a prime number and does not share any factors with 220, the fraction is already in its simplest form. Simplifying fractions is key to ensuring your answer is as clear and concise as possible.