Problem 14 - Elementary Algebra

Question

Perform the indicated multiplications and divisions and express your answers in simplest form. $$\frac{24 a b^{2}}{25 b} \div \frac{-12 a b}{15 a^{2}}$$

Step-by-Step Solution

Answer

\(-\frac{6a^2b}{5}\)

Step 1: Reciprocate the Divisor

When dividing by a fraction, reciprocate (flip) the divisor and change the division sign to multiplication. Thus, \( \frac{24ab^2}{25b} \div \frac{-12ab}{15a^2} \) becomes \( \frac{24ab^2}{25b} \times \frac{15a^2}{-12ab} \).

Step 2: Multiply the Fractions

Multiply the numerators and the denominators. So, \( \frac{24ab^2 \times 15a^2}{25b \times -12ab} \) simplifies to \( \frac{360a^3b^2}{-300ab} \).

Step 3: Simplify the Fraction

To simplify \( \frac{360a^3b^2}{-300ab} \), divide both the numerator and the denominator by their greatest common factor. The GCF here is \( 60ab \). Simplifying, we get \( \frac{6a^2b}{-5} \).

Step 4: Express in Simplest Form

Factor out negative sign from the denominator and rewrite the expression as \( -\frac{6a^2b}{5} \). This is the simplest form.

Key Concepts

Multiplication and Division of Fractions

When working with fractions in algebra, it's important to know how to multiply and divide them correctly. The steps for these operations are straightforward, yet essential to master.

To multiply fractions, you simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. For instance, if you have two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), then their product is \( \frac{a \times c}{b \times d} \).

On the other hand, dividing fractions involves an extra step. When you divide one fraction by another, you multiply the first fraction by the reciprocal of the second one.

• Reciprocating a fraction means flipping its numerator and denominator.
• For example, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \).

After finding the reciprocal of the divisor, proceed as if you're multiplying fractions. Remember to multiply once you've flipped the divisor.

Finding the Greatest Common Factor

Simplifying algebraic fractions often requires finding the greatest common factor (GCF) of the numbers or variables involved. The GCF is the largest number, or expression, that divides each term without a remainder.

To identify the GCF of terms, list the factors of each term and find the largest factor common to both.

• Consider numbers, coefficients, and variables separately.
• For instance, with terms like \(360a^3b^2\) and \(-300ab\), list their factors.
• For \(360\) and \(-300\), the GCF is \(60\).
• Among variables, the lowest powers define the GCF, so \( a^1b^1 \).

Therefore, the GCF of these terms is \(60ab\). Use the GCF to simplify the fraction by dividing both the numerator and denominator by it.

Reciprocal and Division of Fractions

A reciprocal is what you multiply a fraction by to get a result of 1. Simply put, flipping a fraction gives you its reciprocal. This concept is pivotal in dividing fractions since it converts division into a multiplication problem.

When faced with a division problem, such as \( \frac{24ab^2}{25b} \div \frac{-12ab}{15a^2} \), apply the reciprocal and division rule: * Reciprocate the divisor: Flip \( \frac{-12ab}{15a^2} \) to \( \frac{15a^2}{-12ab} \). * Change division to multiplication: Now, \( \frac{24ab^2}{25b} \times \frac{15a^2}{-12ab} \).

This maneuver simplifies division into a multiplication task, and once flipped, you proceed to multiply as usual. After performing the multiplication, remember to simplify the expression further, using techniques such as finding the GCF.