Problem 28 - Algebra 2

Question

Simplify each complex fraction. \(\frac{\frac{2}{x+y}}{\frac{5}{x+y}}\)

Step-by-Step Solution

Answer

The simplified form of the given complex fraction \(\frac{\frac{2}{x+y}}{\frac{5}{x+y}}\) is \(\frac{2}{5}\)

Step 1: Analyze the fraction

First let's see what our fraction looks like. It is a complex (also called compound) fraction that has a numerator of \(\frac{2}{x+y}\) and a denominator of \(\frac{5}{x+y}\). In other words, we are asked to simplify \(\frac{\frac{2}{x+y}}{\frac{5}{x+y}}\)

Step 2: Divide by multiplying with the reciprocal

We all know that division is essentially multiplication with reciprocal of the number (or the fraction in this case) we want to divide by. Therefore, simplify the fractional division by converting it into multiplication with reciprocal of the denominator. In this case, the reciprocal of \(\frac{5}{x+y}\) is \(\frac{x+y}{5}\). The fraction then becomes \(\frac{2}{x+y} * \frac{x+y}{5}\)

Step 3: Cancel out common factors

In the fraction \(\frac{2}{x+y} * \frac{x+y}{5}\), (x+y) is a common factor in the numerator and the denominator and they cancel out each other which leads to 2/5

Key Concepts

Simplification

Simplifying complex fractions can often feel confusing at first, but it all comes down to breaking the problem into smaller, more manageable steps. A complex fraction is essentially a fraction where either the numerator, the denominator, or both, are also fractions.

To simplify a complex fraction, follow these steps:

• First, analyze the given complex fraction and identify the smaller fractions within it. For example, in our exercise, we have a complex fraction: \(\frac{\frac{2}{x+y}}{\frac{5}{x+y}}\).
• Next, simplify this complex fraction by finding a simpler expression. The aim is to turn the complex structure into a single fraction that is easier to handle.

By simplifying a complex fraction, you prepare it for operations like multiplication or addition with other fractions, making your calculations much easier to perform.

Reciprocal

The concept of the reciprocal plays a pivotal role in simplifying complex fractions. The reciprocal of a number or a fraction is essentially what you multiply by to get 1. In other words, the reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\).

In the context of complex fractions, you often need to divide by another fraction. A clever trick is to multiply by the reciprocal instead of doing division directly.

• This idea was applied in our example where we had \(\frac{\frac{2}{x+y}}{\frac{5}{x+y}}\). Instead of dividing by \(\frac{5}{x+y}\), we multiplied by its reciprocal \(\frac{x+y}{5}\).

Understanding the reciprocal is essential as it forms the backbone of fraction division, turning a potentially tricky task into a straightforward multiplication problem.

Fraction Multiplication

Multiplication of fractions is a fundamental concept that is essential for manipulating and simplifying more complex expressions, such as complex fractions. Once you have turned division into multiplying by a reciprocal, you apply fraction multiplication.

The rule for multiplying fractions is simple:

• Multiply the numerators with each other, and then multiply the denominators.

For example, in our complex fraction, after taking the reciprocal, we end up with \(\frac{2}{x+y} \times \frac{x+y}{5}\).
Multiplying these fractions involves:

• Numerators: \(2 \times (x+y)\)
• Denominators: \((x+y) \times 5\)

In this specific exercise, notice that both the numerator and the denominator contain \(x+y\), which cancels out, simplifying our final answer to \(\frac{2}{5}\).

Understanding how to multiply fractions ensures that you can tackle complex algebraic expressions with confidence, leading to clean and simplified results.