Problem 11 - Elementary Algebra

Question

Perform the indicated multiplications and divisions and express your answers in simplest form. $$\frac{3 a^{2}}{7} \div \frac{6 a}{28}$$

Step-by-Step Solution

Answer

The simplest form is \( 2a \).

Step 1: Interpret the Problem

We are given the expression \( \frac{3a^2}{7} \div \frac{6a}{28} \). Our task is to perform the division and express the result in the simplest form.

Step 2: Convert Division to Multiplication

When dividing fractions, you multiply the first fraction by the reciprocal of the second fraction. Therefore, \( \frac{3a^2}{7} \div \frac{6a}{28} \) becomes \( \frac{3a^2}{7} \times \frac{28}{6a} \).

Step 3: Multiply the Fractions

Multiply the numerators and denominators: \( \frac{3a^2 \times 28}{7 \times 6a} = \frac{84a^2}{42a} \).

Step 4: Simplify the Resulting Fraction

To simplify, divide the numerator and the denominator by their greatest common divisor. The GCD of 84 and 42 is 42, and note that \( a \) cancels one \( a \) from the numerator. Thus, \( \frac{84a^2}{42a} \) simplifies to \( \frac{2a}{1} \) or simply \( 2a \).

Key Concepts

Fraction Division

When it comes to dividing fractions, a simple trick is to use multiplication instead. The process involves multiplying the first fraction by the reciprocal of the second.
During the division of fractions, remember these steps:

• Identify the two fractions involved.
• Find the reciprocal of the second fraction. The reciprocal is essentially flipping the numerator and the denominator.
• Multiply the first fraction by this new reciprocal fraction.

Let's see this in action with an example:
Given the expression \( \frac{3a^2}{7} \div \frac{6a}{28} \), we convert it to multiplication by using the reciprocal: \( \frac{3a^2}{7} \times \frac{28}{6a} \) and proceed from there.
Utilizing the reciprocal method helps transition from a potentially challenging division problem to a more straightforward multiplication task.

Algebraic Expressions

Algebraic expressions like \( \frac{3a^2}{7} \) or \( \frac{6a}{28} \) may seem daunting, but they consist of numbers, variables, and operations. Understanding these expressions involves recognizing their components:

• Numerators and denominators: In fractions, these represent the parts of division, with the numerator divided by the denominator.
• Variables like \( a \): These are symbols that can represent numbers and often come with exponents like \( a^2 \).
• Operations: In this example, operations include division of fractions and multiplication.

When tackling algebraic expressions, it's vital to systematically handle each part.
Start by understanding the variables' roles and the mathematical operations that link all parts together.
This understanding ensures we approach any simplification or operations correctly, as seen in converting division in the problem to multiplication.

Simplifying Fractions

Once we have the new expression from our multiplication, the next step is simplifying the fractions.
This involves reducing fractions to their simplest form, a crucial skill where you strip down fractions to their most basic representation.

• Determine the greatest common divisor (GCD) of both the numerator and the denominator.
• Divide both parts by the GCD to simplify the fraction.
• Cancel out like terms if working with algebraic terms, such as removing common factors like \( a \) from both numerator and denominator.

In the example, \( \frac{84a^2}{42a} \) simplifies to \( \frac{2a}{1} \) using these steps. We find the GCD of 84 and 42, which is 42, and simplify by cancelling equivalent terms.
This results in the beautifully simple expression, \( 2a \), showcasing how powerful simplification can be.