Problem 85 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Write the answer as a fraction or mixed number in simplest form. \(\frac{3}{4}+\frac{5}{8}-\frac{1}{2}\)

Step-by-Step Solution

Answer

\(\frac{7}{8}\)

Step 1: Identify common denominator

Identify a common denominator for all the fractions. In this case, the common denominator for 4, 8, and 2, is 8.

Step 2: Convert fractions to have common denominator

Convert all fractions to have this common denominator: \(\frac{3}{4}=\frac{6}{8}\), \( \frac{5}{8}=\frac{5}{8}\), \(\frac{1}{2}=\frac{4}{8}\)

Step 3: Perform the operation

Now, perform the operation of addition and subtraction: \( \frac{6}{8} + \frac{5}{8} - \frac{4}{8} = \frac{7}{8}\)

Step 4: Final simplification

Here, the result is already in simplest form, as 7 and 8 have no common factors other than 1. So, our final answer is \(\frac{7}{8}\)

Key Concepts

Common Denominator

Before you can add or subtract fractions, they must share the same denominator. This is because the denominator, which is the bottom number in a fraction, represents the total number of equal parts. To have a common ground to perform arithmetic operations, we need all fractions involved to reference these same parts. Finding a common denominator involves identifying the least common multiple (LCM) of the denominators involved.

Consider the fractions \(\frac{3}{4}, \frac{5}{8},\) and \(\frac{1}{2}\). The denominators are 4, 8, and 2. The smallest number divisible by all these denominators is 8, making it our common denominator. By converting each fraction to an equivalent one with the denominator of 8, we can seamlessly proceed to the next step of the operation.

Simplifying Fractions

Simplifying fractions is the process of reducing the fraction to its simplest form. To simplify a fraction, you divide both the numerator and the denominator by their greatest common factor (GCF).

In the exercise given, however, you begin by converting denominators: \(\frac{3}{4} = \frac{6}{8},\) \(\frac{5}{8},\) and \(\frac{1}{2} = \frac{4}{8}\). These conversions set the stage for adding and subtracting the fractions correctly.

After performing the operations and obtaining \(\frac{7}{8}\), we check whether the fraction can be simplified further. The GCF of 7 and 8 is 1, meaning it's already in its simplest form. When a fraction is simplified, the process involves ensuring no common factors remain other than 1. Keeping fractions in their simplest form is crucial for clarity and ease of understanding.

Addition and Subtraction of Fractions

Once fractions share a common denominator, you can easily perform addition and subtraction. Simply add or subtract the numerators while keeping the same denominator. It's like counting how many parts you have when they are all the same size.

In the exercise, after conversion, you have: \(\frac{6}{8} + \frac{5}{8} - \frac{4}{8}\).
First, perform the addition: \(6 + 5 = 11\). You now have \(\frac{11}{8}\) from those two fractions. Next, subtract the remaining fraction: \(11 - 4 = 7\), giving you \(\frac{7}{8}\).

This approach requires accuracy in adjusting numerators during the process and ensures that the relationships between parts are maintained. It's an essential skill for handling fractions with different denominators in everyday math.