Problem 39 - Prealgebra

Question

Find each product. Write in simplest form. $$\frac{3 x}{y} \cdot \frac{9 y}{x}$$

Step-by-Step Solution

Answer

The simplest form of the product is 27.

Step 1: Multiply Numerators

First, multiply the numerators of the given fractions. You have \(3x\) and \(9y\). The product of the numerators is \(3x \times 9y = 27xy\).

Step 2: Multiply Denominators

Next, multiply the denominators of the given fractions. You have \(y\) and \(x\). The product of the denominators is \(y \times x = xy\).

Step 3: Formulate the Fraction

Now, place the result of the products of the numerators over the product of the denominators to form a new fraction. Thus, you have \(\frac{27xy}{xy}\).

Step 4: Simplify the Fraction

Cancel the common terms in the numerator and denominator. Both \(27xy\) and \(xy\) have the common factors \(xy\). After canceling, you are left with \(27\).

Key Concepts

Simplifying Fractions

Simplifying fractions is the process of reducing fractions to their simplest form. This means ensuring the numerator and the denominator no longer have any common factors except for 1. Simplification helps in making fractions easier to understand and work with.

• Identify any common factors between the numerator and the denominator.
• Divide both by their greatest common factor (GCF) to get a simplified fraction.

In our exercise, the fraction \(\frac{27xy}{xy}\) was simplified by noticing both numerator and denominator had \(xy\) as a factor. Thus, dividing both by \(xy\) left us with just 27.

Numerator and Denominator

Fractions are composed of two parts: the numerator and the denominator. Understanding these is very crucial:

• The numerator is the top number in the fraction, representing how many parts are being considered.
• The denominator is the bottom number, signifying the total number of equal parts the whole is divided into.

In \(\frac{3x}{y} \cdot \frac{9y}{x}\), the numerators are \(3x\) and \(9y\), and the denominators are \(y\) and \(x\).
Recognizing these helps in multiplying fractions correctly and simplifying them effectively.

Algebraic Expressions

Algebraic expressions involve variables, coefficients, and constants. When dealing with them in fractions, it's important to perform operations accurately.

• Variables like \(x\) and \(y\) can represent unknown values.
• Coefficients are numbers multiplying the variables.
• Understood well, these form the foundation for operations like multiplication.

Algebraic multiplication is the same as numerical multiplication but keep track of each term. In this exercise: \(3x\) multiplied by \(9y\) gives \(27xy\), demonstrating how variables combine, while still relying on multiplication rules.

Step by Step Solution

Providing a step-by-step solution makes the process of multiplication and simplification clear and easy to follow. Here's a breakdown of the provided solution:
First, identify and multiply the numerators: \(3x\) and \(9y\) gives \(27xy\). Then select and multiply the denominators: \(y\) by \(x\) results in \(xy\).
Form the new fraction \(\frac{27xy}{xy}\). Notice that the common factor \(xy\) appears in both the numerator and the denominator. Cancel out these common terms to simplify the fraction fully to 27.
This methodical approach ensures each aspect of fraction multiplication is covered and demonstrated for understanding and application in similar problems.