Problem 24 - Algebra 2

Question

Simplify each complex fraction. \(\frac{\frac{2}{x+y}}{3}\)

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \(\frac{2}{3*(x+y)}\)

Step 1: Identify the reciprocal of denominator

The reciprocal of a number is 1 divided by that number. Therefore, the reciprocal of 3 is 1/3.

Step 2: Multiply by the reciprocal

Multiplying the entire fraction by the reciprocal of the denominator, it looks like: \(\frac{2}{x+y} * \frac{1}{3}\).

Step 3: Simplify the fraction

When multiplying fractions, multiply the numerators together for the new numerator and the denominators together for the new denominator. Therefore, \(\frac{2}{x+y} * \frac{1}{3} = \frac{2*1}{(x+y)*3}\).

Key Concepts

Reciprocal

When dealing with complex fractions, understanding the concept of a reciprocal is crucial. A reciprocal is essentially flipping a fraction upside down. This means that if you have a number, say 3, its reciprocal would be \( \frac{1}{3} \). This concept is invaluable when simplifying complex fractions because it allows you to transform division into multiplication.
Using the reciprocal of the denominator, you can further simplify the fraction and solve the problem more efficiently.
Applying this method of using reciprocals to complex fractions is a fundamental step leading to an easier solution.

Simplification

Simplification is the process of making an expression easier to understand or work with. When dealing with complex fractions like \( \frac{\frac{2}{x+y}}{3} \), simplification is key to finding the answer efficiently.
Simplifying involves breaking down the problem into simpler, more manageable parts. In our example, this means using the reciprocal of the denominator to change the division into multiplication.
After this, combine the expressions through multiplication and reduce them to their simplest form. Simplification not only helps reach the answer more quickly but also makes the process more transparent and easier to follow.

Multiplying Fractions

Multiplying fractions is a straightforward but essential operation in algebra. After replacing the division with a reciprocal, you multiply fractions to find the solution. For the expression \( \frac{2}{x+y} \times \frac{1}{3} \), the process involves multiplying the numerators and denominators separately.

• First, multiply the numerators: \( 2 \times 1 = 2 \).
• Next, multiply the denominators: \( (x+y) \times 3 = 3(x+y) \).

This results in a new fraction: \( \frac{2}{3(x+y)} \).
Multiplying fractions is straightforward when you approach it step by step, making it less intimidating and more manageable.