Problem 47 - Elementary Algebra

Question

For Problems 41-60, simplify each of the complex fractions. $$ \frac{\frac{3}{x}}{\frac{9}{y}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{y}{3x} \).

Step 1: Understand the structure of the complex fraction

The given expression \( \frac{\frac{3}{x}}{\frac{9}{y}} \) is a complex fraction, meaning it has fractions both in the numerator and the denominator. Our goal is to simplify this into a single fraction.

Step 2: Rewrite the complex fraction as a division problem

A complex fraction can be rewritten using division. This means \( \frac{\frac{3}{x}}{\frac{9}{y}} \) can be expressed as \( \frac{3}{x} \div \frac{9}{y} \).

Step 3: Apply the division rule for fractions

To divide by a fraction, multiply by its reciprocal. Thus, we rewrite \( \frac{3}{x} \div \frac{9}{y} \) as \( \frac{3}{x} \times \frac{y}{9} \).

Step 4: Multiply the fractions

To multiply two fractions, multiply their numerators and their denominators: \( \frac{3}{x} \times \frac{y}{9} = \frac{3y}{9x} \).

Step 5: Simplify the resulting fraction

Simplify \( \frac{3y}{9x} \) by cancelling common factors. The numerator and the denominator can both be divided by 3: \( \frac{3y}{9x} = \frac{y}{3x} \).

Key Concepts

Fractions

Fractions are a fundamental concept in mathematics, representing a part of a whole. A fraction is composed of two parts: a numerator and a denominator. The numerator is what counts, or the top number, while the denominator is what divides, or the bottom number. For example, in the fraction \( \frac{3}{4} \), 3 is the numerator and 4 is the denominator.
Breaking down fractions is a key skill, especially when dealing with complex fractions. Complex fractions, like the one in your exercise, are fractions where either the numerator, the denominator, or both, are themselves fractions.

• Numerators: The number of parts considered.
• Denominators: The total number of equal parts the whole is divided into.
• Complex fractions: Fractions inside a fraction.

Simplifying Fractions

Simplifying fractions means making the fraction as simple as possible. We achieve this by dividing both the numerator and the denominator by their greatest common divisor. This makes calculations easier and results clearer.
When simplifying complex fractions, the goal is to convert them into a simpler form. For example, the complex fraction \( \frac{\frac{3}{x}}{\frac{9}{y}} \) is simplified by following the steps:

• Convert division into multiplication by the reciprocal.
• Multiply straight across the numerators and denominators.
• Reduce the resulting fraction by finding common factors.

The simplified fraction \( \frac{y}{3x} \) is much easier to interpret and use in further calculations.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It often involves finding the unknown or assigning letters to numbers. It provides a way to think logically about numbers and how they relate to one another.
In the context of simplifying fractions, algebraic techniques can be used to handle variable expressions within fractions. Variables like \( x \) and \( y \) allow us to work with generic numbers, leading to more inclusive solutions.
Using algebra, we can simplify expressions and solve equations involving fractions by applying properties such as the distributive, associative, and commutative laws.

Reciprocal

The reciprocal of a number is what you multiply with that number to get one. For any nonzero number \( a \), its reciprocal is \( \frac{1}{a} \).
When working with fractions, especially when dividing them like in your exercise, reciprocals become very important. Division by a fraction is the same as multiplication by its reciprocal. This concept is crucial for simplifying complex fractions.

• To find the reciprocal of \( \frac{9}{y} \), flip it: it becomes \( \frac{y}{9} \).
• Division by \( \frac{9}{y} \) turns into multiplication by \( \frac{y}{9} \).
• Understanding reciprocals simplifies many algebraic manipulations, especially when working with fractions.