Problem 10 - Elementary Algebra

Question

Perform the indicated multiplications and divisions and express your answers in simplest form. $$\left(\frac{4 a b}{10}\right)\left(-\frac{30 a}{22 b}\right)$$

Step-by-Step Solution

Answer

\( \frac{-6a^2}{11} \)

Step 1: Understand the Problem

We need to multiply the two fractions \( \left(\frac{4ab}{10}\right) \) and \( \left(-\frac{30a}{22b}\right) \). After multiplication, we simplify the resulting expression to its simplest form.

Step 2: Multiply the Numerators and Denominators

Multiply the numerators: \(4ab \times (-30a) = -120a^2b\). Multiply the denominators: \(10 \times 22b = 220b\). The resulting fraction is \(\frac{-120a^2b}{220b}\).

Step 3: Simplify the Fraction

First, notice that \(b\) is common in both the numerator and the denominator. Cancel \(b\) from both to get \(\frac{-120a^2}{220}\).

Step 4: Reduce the Fraction

Find the greatest common divisor (GCD) of 120 and 220, which is 20. Divide both the numerator and the denominator by 20: \(\frac{-120a^2}{220} = \frac{-6a^2}{11}\).

Step 5: Final Simplified Form

The fraction \(\frac{-6a^2}{11}\) is in its simplest form since 6 and 11 have no common factors other than 1.

Key Concepts

Multiplication of Fractions

Multiplying fractions is all about working with numerators and denominators. A fraction is made up of a numerator (the top number) and a denominator (the bottom number). To multiply two fractions, you simply:

• Multiply the numerators together to get the new numerator.
• Multiply the denominators together to get the new denominator.

In this exercise, we're multiplying \( \frac{4ab}{10} \) and \( -\frac{30a}{22b} \). To do this:

• Multiply the numerators: \(4ab \times (-30a) = -120a^2b\).
• Multiply the denominators: \(10 \times 22b = 220b\).

This results in the new fraction \( \frac{-120a^2b}{220b} \). The key point to remember is to multiply across the numerators and denominators, keeping track of any negative signs, as shown here. This sets the stage for the next step: simplification.

Algebraic Expressions

Algebraic expressions can add an extra layer to problem-solving because they contain variables like \(a\) and \(b\). When multiplying algebraic fractions, remember:

• Each variable acts as a placeholder for a number.
• Follow the same rules you use for numbers when multiplying.
• If the same variable appears in both the numerator and the denominator, you can often simplify by canceling out the common factors.

In our exercise, we had \(a\) and \(b\) as parts of an algebraic fraction. After multiplying the fractions:

• The expression \(-120a^2b\) was in the numerator.
• The expression \(220b\) was in the denominator.

By recognizing that \(b\) is in both the numerator and the denominator, you can simplify by canceling \(b\), leading to \(\frac{-120a^2}{220}\). Keep in mind that simplifying algebraic expressions relies on identifying and removing these common factors.

Simplifying Fractions

Simplifying fractions is all about making them as simple as possible—using the smallest possible numbers. Here's how you can simplify:

• Identify any common factors in the numerator and the denominator.
• Calculate the greatest common divisor (GCD).
• Divide both the numerator and the denominator by the GCD.

For the exercise, we had to simplify \( \frac{-120a^2}{220} \). Both 120 and 220 have common factors, and their greatest common divisor (GCD) is 20. Dividing the numerator and the denominator by 20 gives us \( \frac{-6a^2}{11} \). By ensuring there are no common factors left (other than 1), you confirm that the fraction is fully simplified.
Practicing this process helps you spot opportunities to simplify quickly, leading to cleaner, more manageable results in algebraic expressions and beyond.