Problem 23 - Prealgebra

Question

Find the quotient in each case by replacing the divisor by its reciprocal and multiplying. $$\frac{x^{2}}{y^{3}} \div \frac{x}{y^{2}}$$

Step-by-Step Solution

Answer

The quotient is \( \frac{x}{y} \).

Step 1: Identify the problem

We need to divide the fraction \( \frac{x^{2}}{y^{3}} \) by \( \frac{x}{y^{2}} \). The division of fractions can be converted into multiplication by taking the reciprocal of the divisor.

Step 2: Write the reciprocal of the divisor

The divisor is \( \frac{x}{y^{2}} \). Its reciprocal is \( \frac{y^{2}}{x} \).

Step 3: Convert division to multiplication

Replace the division operation with multiplication by multiplying \( \frac{x^{2}}{y^{3}} \) with the reciprocal of the divisor: \( \frac{x^{2}}{y^{3}} \times \frac{y^{2}}{x} \).

Step 4: Perform the multiplication

Multiply the numerators together and the denominators together. This gives us: \( \frac{x^{2} \cdot y^{2}}{y^{3} \cdot x} \).

Step 5: Simplify the expression

Simplify the expression by canceling common factors: cancel \( x \) from both the numerator and the denominator, and simplify \( y^{2}/y^{3} \): \[ \frac{x^{2} \cdot y^{2}}{y^{3} \cdot x} = \frac{x \cdot y^{2}}{y^{3}} = \frac{x}{y} \].

Key Concepts

Reciprocal

The reciprocal is a fundamental concept in fraction division. Simply put, the reciprocal of a fraction is what you get when you flip the fraction upside down. For example, if you have the fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). In the exercise we discussed, the divisor is \( \frac{x}{y^{2}} \). To find its reciprocal, we flip it to get \( \frac{y^{2}}{x} \).Using the reciprocal transforms division into multiplication, simplifying the process while maintaining the mathematical value. This is essential because multiplying fractions is a much more straightforward procedure than direct division. Thus, finding the reciprocal is the first step in division involving fractions. It sets the stage for replacing the division operation with multiplication, thereby transforming the problem into a more manageable form.

Simplifying Fractions

Simplifying fractions is all about making fractions as simple as possible to understand and work with. When a fraction is simplified, it is expressed in its lowest terms, meaning the numerator and the denominator no longer share any common factors except for 1.In our exercise, after multiplying the fractions, we arrive at \( \frac{x^{2} \cdot y^{2}}{y^{3} \cdot x} \). To simplify, we first look for common factors in the numerator and the denominator:

• The factor \( x \) appears in both the numerator and the denominator. We can cancel one of them out, reducing \( x^{2} \) to \( x \) and eliminating \( x \) from the denominator completely.
• The \( y \'s \) in the numerator and the denominator can also simplify. The term \( \frac{y^{2}}{y^{3}} \) becomes \( \frac{1}{y} \) after cancelling \( y^{2} \).

Ultimately, simplifying fractions is a crucial step because it prepares the result for further calculations or interpretations in the simplest, most efficient form.

Multiplication of Fractions

When dealing with multiplication of fractions, you multiply the numerators together and the denominators together. This is a straightforward procedure than can be easily applied.In our exercise, once we take the reciprocal of the divisor and replace the division sign with multiplication, we have:\[\frac{x^{2}}{y^{3}} \times \frac{y^{2}}{x}\]Here's how you perform the multiplication:

• Multiply the numerators: \( x^{2} \times y^{2} \).
• Multiply the denominators: \( y^{3} \times x \).

The resulting fraction, \( \frac{x^{2} \cdot y^{2}}{y^{3} \cdot x} \), is what you get before you proceed to simplify it.Mastering the multiplication of fractions can boost confidence in handling complex problems. It is often easier than other operations, so getting comfortable with it is key to efficiently solving fraction problems.