Problem 44 - Algebra and Trigonometry

Question

Perform the multiplication or division and simplify. \(\frac{x}{y / z}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{xz}{y} \).

Step 1: Understand the Problem

We need to simplify the expression \( \frac{x}{y / z} \). The expression has a fraction in the denominator, which we need to simplify by performing a division.

Step 2: Reciprocate the Fraction

The division sign in the denominator \( y/z \) suggests we should take the reciprocal to simplify the fraction. The reciprocal of \( y/z \) is \( z/y \).

Step 3: Multiply Instead of Divide

Replace the division by multiplication using the reciprocal. Therefore, \( \frac{x}{y / z} \) becomes \( x \times \frac{z}{y} \).

Step 4: Simplify the Expression

Now we perform the multiplication: \( x \times \frac{z}{y} = \frac{xz}{y} \). This expression is the product of \( x \) and \( \frac{z}{y} \).

Step 5: Final Simplification Check

Check if there are any common factors in the numerator and the denominator that can be simplified. Here, since \( x, y, \) and \( z \) are variables, there is no further simplification possible unless specific values are given.

Key Concepts

Multiplication and Division of Fractions

When working with fractions, it's essential to understand how multiplication and division operate. To multiply fractions, you multiply the numerators together and the denominators together. Thus, you have a straightforward operation. For example, to multiply \( \frac{a}{b} \) and \( \frac{c}{d} \), you would perform:

• Numerator: \( a \times c \)
• Denominator: \( b \times d \)

This results in the fraction \( \frac{ac}{bd} \).
When dividing fractions, you need to flip, or reciprocate, the second fraction and then multiply. This is often easier since multiplication deals with direct operations rather than changing values. For instance, dividing \( \frac{a}{b} \) by \( \frac{c}{d} \) involves multiplying \( \frac{a}{b} \) by the reciprocal of \( \frac{c}{d} \), which is \( \frac{d}{c} \). This simplifies the operation to multiplication:

• \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc} \)

It's crucial to apply the reciprocal correctly to ensure the division is accurate and simplified where possible.

Reciprocal of a Fraction

The reciprocal of a fraction is a fundamental concept that involves flipping the fraction upside down. For a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). This switch of numerator and denominator is vital because it converts division into multiplication.
Using the reciprocal is particularly helpful when dealing with complex fractions or expressions that require division. The expression \( \frac{x}{y / z} \), for instance, involves taking the reciprocal of \( \frac{y}{z} \) to convert it into a multiplication problem.
After replacing \( \frac{y}{z} \) with its reciprocal, \( \frac{z}{y} \), you can easily multiply:

• Expression becomes \( x \times \frac{z}{y} \)
• This is much simpler and avoids unnecessary division complications.

Remember, the reciprocal changes the operation from division to multiplication while keeping the expression equivalent.

Algebraic Expression Simplification

Simplifying algebraic expressions involves reducing an expression to its simplest form. This process includes performing operations and combining like terms to make the expression easier to work with. When dealing with fractions, simplification often means breaking down the fraction to the lowest terms.
For the expression \( x \times \frac{z}{y} \), you multiply to obtain \( \frac{xz}{y} \). The next step in simplification is to check for any common factors that might exist between the numerator and the denominator. If variables like \( x \), \( y \), and \( z \) were actual numbers, you'd factor them to find any potential simplifications.

• For example: If \( x = 4 \), \( z = 2 \), and \( y = 8 \), then \( \frac{4 \times 2}{8} = \frac{8}{8} = 1 \).
• This example shows how common factors can simplify expressions to fewer terms.

When only variables are present without numerical values, further simplification depends on a clear understanding of algebraic rules. It's also essential to ensure that any simplification remains valid across any potential values for the variables involved.