Problem 42 - Prealgebra

Question

Multiply or divide as indicated. $$\frac{a^{2} b}{c^{3}} \div \frac{a b^{2}}{c^{3}}$$

Step-by-Step Solution

Answer

The result of the expression is \( \frac{a}{b} \).

Step 1: Rewrite the Division as Multiplication

When dividing by a fraction, you multiply by its reciprocal. Therefore, \( \frac{a^{2} b}{c^{3}} \div \frac{a b^{2}}{c^{3}} \) is equivalent to \( \frac{a^{2} b}{c^{3}} \cdot \frac{c^{3}}{a b^{2}} \).

Step 2: Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together: \( \frac{a^{2} b \cdot c^{3}}{c^{3} \cdot a b^{2}} \). This simplifies to \( \frac{a^{2} b c^{3}}{c^{3} a b^{2}} \).

Step 3: Simplify the Fraction

Cancel out the common terms in the numerator and the denominator. Both terms have \(c^3\), \(a\), and \(b\), so the expression simplifies to \( \frac{a}{b} \).

Key Concepts

Division of Fractions

When you see a division problem involving fractions, it might seem tricky at first. But there's a simple rule that makes it much easier: dividing by a fraction is the same as multiplying by its reciprocal. This means that when you have an expression that involves division, you can convert it into a multiplication problem.

• Instead of dividing by a fraction, flip the fraction upside down and multiply.
• For example, if you have \( \frac{a}{b} \div \frac{c}{d} \), you can rewrite it as \( \frac{a}{b} \times \frac{d}{c} \).
• This step simplifies the problem and makes it easier to solve.

Applying this to our original exercise, \( \frac{a^{2} b}{c^{3}} \div \frac{a b^2}{c^3} \) becomes \( \frac{a^{2} b}{c^{3}} \cdot \frac{c^3}{a b^2} \). Now, we can proceed with multiplication, which is a simpler operation.

Simplifying Fractions

Once you have rewritten the division as multiplication, the next step is to simplify the fractions. Simplifying fractions involves breaking down both the numerator and the denominator into their smallest parts to cancel out any common factors.

• Write both the top and bottom of the fraction as a product of their factors.
• Cancel out the factors that are common in both the numerator and the denominator.
• For example, if both have the factor \( x \), you can simplify \( \frac{xy}{xz} \) to \( \frac{y}{z} \).

In our specific exercise, after rewriting the division as multiplication, we have the fraction \( \frac{a^{2} b c^{3}}{c^{3} a b^2} \). Notice that \( c^3 \), \( a \), and \( b \) appear in both parts of the fraction. These can be canceled out, leaving us with a simplified result.

Reciprocal in Division

Understanding the reciprocal is key when you're working with fraction division. The reciprocal of a number is simply one divided by that number. For example, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \). This concept allows you to flip the fraction and switch the division into multiplication, making the solution much simpler.

• To find the reciprocal, swap the numerator and denominator.
• In the context of division, convert \( \frac{a}{b} \div \frac{c}{d} \) into \( \frac{a}{b} \times \frac{d}{c} \).
• This approach simplifies complex division into more manageable multiplication.

In the original problem, we used the reciprocal of \( \frac{a b^2}{c^3} \) which is \( \frac{c^3}{a b^2} \), and thus turned a division problem into a multiplication problem, simplifying the overall process. Understanding and applying the concept of reciprocals is crucial for solving fraction division problems efficiently.