Problem 22 - Algebra 2

Question

Simplify each complex fraction. \(\frac{\frac{1}{x}}{\frac{2}{y}}\)

Step-by-Step Solution

Answer

The simplified form of the complex fraction \(\frac{\frac{1}{x}}{\frac{2}{y}}\) is \(\frac{y}{2x}\).

Step 1: Understand the Problem

The given complex fraction is \(\frac{\frac{1}{x}}{\frac{2}{y}}\). This can also be read as \(\frac{1}{x}\) divided by \(\frac{2}{y}\). We need to simplify this fraction to its lowest terms.

Step 2: Use the Rule of Reciprocal

Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we rewrite the given complex fraction as \(\frac{1}{x} * \frac{y}{2}\) by flipping the fraction \(\frac{2}{y}\).

Step 3: Multiply the Fractions

We then perform the multiplication. In multiplying fractions, we multiply the numerators and the denominators separately. This yields: \(\frac{1*y}{x*2} = \frac{y}{2x}\).

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form. This is done by removing any common factors in the numerator and the denominator, essentially making the fraction as "short" or "compact" as possible. To understand simplification, think of it as tidying up! When dealing with complex fractions like the one in the exercise, consider it as an opportunity to practice your fraction skills. A complex fraction is a fraction where the numerator, the denominator, or both are also fractions. The first step in simplifying such fractions is to recognize the fractions within the fraction.

• First, understand the problem — identify the inner fractions present in the complex fraction.
• Next, deal with the operation involved, which usually requires finding the reciprocal to simplify.
• Finally, ensure that the resulting fraction is reduced to its simplest terms.

Breaking down the fraction into manageable steps makes it easier to simplify. With practice, even the most complex fractions will become simple to handle!

Reciprocal

The concept of a reciprocal is central to simplifying complex fractions. A reciprocal of a number or fraction is simply what you multiply that number by to get 1. For fractions, the reciprocal is found by swapping the numerator and the denominator.For example, if you have a fraction like \( \frac{2}{3} \), its reciprocal is \( \frac{3}{2} \). Similarly, the reciprocal of \( \frac{1}{x} \) would be \( x \). It's that straightforward! This concept is crucial for dividing fractions because, instead of performing a division, you can convert the division into a multiplication problem by multiplying by the reciprocal. In our exercise, the reciprocal transforms \( \frac{2}{y} \) into \( \frac{y}{2} \), which makes the subsequent multiplication straightforward and effective.

• Always ensure you correctly identify the reciprocals.
• Use the reciprocal to change division operations into multiplication.
• This technique allows for simplifying complex operations into simpler ones.

By mastering reciprocals, you'll simplify fractions with ease and confidence.

Basic Algebra Operations

Understanding basic algebra operations is key to handling fractions and complex fractions efficiently. These operations include addition, subtraction, multiplication, and division, each of which has specific rules when applied to fractions.For multiplication, when you're combining fractions, you multiply the numerators together to find the new numerator, and the denominators together to find the new denominator. Simple as that! For example, multiplying \( \frac{1}{x} \) and \( \frac{y}{2} \) gives you \( \frac{1\cdot y}{x\cdot 2} = \frac{y}{2x} \). Pay close attention to numerators and denominators during these operations. When dealing with fractions, remember:

• Add or subtract fractions with the same denominator directly, altering only the numerators.
• Find a common denominator for addition or subtraction with differing denominators.
• Perform multiplication and division by flipping/reciprocating and then multiplying.

These basic operations become second nature with practice, allowing you to tackle more complex algebraic tasks with ease!