Problem 122 - Beginning and Intermediate Algebra

Question

Write a numerical expression for each phrase, and simplify the expression. The sum of \(\frac{3}{4}\) and \(\frac{1}{2},\) times the difference of \(\frac{2}{3}\) and \(\frac{1}{6}\)

Step-by-Step Solution

Answer

\(\frac{5}{8}\)

Step 1: Identify the components

The phrase consists of two parts: the sum of \(\frac{3}{4}\) and \(\frac{1}{2}\), and the difference of \(\frac{2}{3}\) and \(\frac{1}{6}\).

Step 2: Calculate the sum

Find the sum of \(\frac{3}{4}\) and \(\frac{1}{2}\). First, convert \(\frac{1}{2}\) to a fraction with the same denominator as \(\frac{3}{4}\): \(\frac{1}{2} = \frac{2}{4}\). Now add the fractions: \(\frac{3}{4} + \frac{2}{4} = \frac{5}{4}\).

Step 3: Calculate the difference

Now, find the difference of \(\frac{2}{3}\) and \(\frac{1}{6}\). First, convert \(\frac{2}{3}\) to a fraction with the same denominator as \(\frac{1}{6}\): \(\frac{2}{3} = \frac{4}{6}\). Then subtract the fractions: \(\frac{4}{6} - \frac{1}{6} = \frac{3}{6} = \frac{1}{2}\).

Step 4: Multiply the results

Multiply the results of the sum and the difference: \(\frac{5}{4} \times \frac{1}{2}\). Multiply the numerators and the denominators: \(\frac{5 \times 1}{4 \times 2} = \frac{5}{8}\).

Key Concepts

Fractions

Fractions are a way to represent a part of a whole number. When we talk about fractions, we use two numbers: the numerator and the denominator.

• The numerator is the top number and represents how many parts we have.
• The denominator is the bottom number and shows the total number of equal parts the whole is divided into.

For example, in the fraction \(\frac{3}{4}\), 3 is the numerator, and 4 is the denominator. This means we have 3 out of 4 equal parts of a whole.
Understanding fractions is essential for tasks such as addition and subtraction of fractions, and multiplication of fractions. Let’s dive into those topics next.

Addition and Subtraction of Fractions

Adding and subtracting fractions may seem tricky at first, but with practice, it becomes quite simple. The key step in both processes is to ensure that the fractions have a common denominator.

• Common Denominator: A common denominator is a shared multiple of the denominators of two or more fractions.

Addition Example: To add \(\frac{3}{4}\) and \(\frac{1}{2}\), we first convert \(\frac{1}{2}\) to a fraction with the same denominator as \(\frac{3}{4}\). Since \(\frac{3}{4}\) has a denominator of 4, we convert \(\frac{1}{2}\) to \(\frac{2}{4}\). Now we can add: \(\frac{3}{4} + \frac{2}{4} = \frac{5}{4} \).

• When adding fractions with the same denominator, just add the numerators and keep the denominator the same.

Subtraction Example: To subtract \(\frac{1}{6}\) from \(\frac{2}{3}\), we convert \(\frac{2}{3}\) to \(\frac{4}{6}\) so that both fractions have the same denominator. Now we subtract: \(\frac{4}{6} - \frac{1}{6} = \frac{3}{6} = \frac{1}{2}\).

• When subtracting fractions with the same denominator, subtract the numerators and keep the denominator the same.

Multiplication of Fractions

Multiplying fractions is more straightforward compared to addition and subtraction. There is no need to find a common denominator. Instead, you multiply the numerators together and the denominators together.
Example: To multiply \(\frac{5}{4}\) by \(\frac{1}{2}\), we simply multiply the numerators and the denominators: \(\frac{5 \times 1}{4 \times 2} = \frac{5}{8}\).
Here are the steps broken down:

• Multiply the numerators: \ 5 \times 1 = 5 \ .
• Multiply the denominators: \ 4 \times 2 = 8 \ .

The final result is \ \frac{5}{8} \ .
This can be particularly useful in algebra when working with expressions that include fractions.