Problem 48 - Elementary and Intermediate Algebra

Question

Divide, and then simplify, if possible. \(\frac{3 x+6}{40} \div \frac{3 x^{2}}{24}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3(x+2)}{5x^2} \).

Step 1: Rewrite the Division as Multiplication

To divide by a fraction, you multiply by its reciprocal. Rewrite the expression as: \[ \frac{3x+6}{40} \div \frac{3x^2}{24} = \frac{3x+6}{40} \times \frac{24}{3x^2} \]

Step 2: Simplify Fractions

Simplify the individual fractions before multiplying. For \( \frac{3x+6}{40} \), factor the numerator: \[ \frac{3(x+2)}{40} \]For \( \frac{24}{3x^2} \), divide numerator and denominator by 3:\[ \frac{8}{x^2} \]

Step 3: Multiply the Fractions

Multiply the simplified fractions: \[ \frac{3(x+2)}{40} \times \frac{8}{x^2} = \frac{3(x+2) \times 8}{40x^2} \]Multiply the numerators and the denominators:\[ \frac{24(x+2)}{40x^2} \]

Step 4: Simplify the Result

Simplify the fraction \( \frac{24(x+2)}{40x^2} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 8:\[ \frac{3(x+2)}{5x^2} \]

Key Concepts

Simplifying Fractions

In algebra, simplifying fractions is about reducing the expression to its simplest form. This means making the numerator and the denominator as small as possible without changing the value of the fraction. To simplify a fraction, follow these main steps:

• Factor the numerator and the denominator. This is like breaking down each number or expression into its basic components. For example, in the fraction \( \frac{3x+6}{40} \), you can factor the numerator as \( 3(x+2) \).
• Cancel common factors. Look for numbers or variables that appear in both the numerator and the denominator and cancel them out. The goal is to make the fraction less cluttered.

A simpler fraction makes it easier to work with the expression and find solutions. Remember, the simpler the expression, the easier it is to understand and manipulate.

Reciprocal Multiplication

When dividing by a fraction, you can multiply by its reciprocal instead. This is because dividing by a number is the same as multiplying by its reciprocal.Here's how it works:

• Create the reciprocal. Flip the fraction you are dividing by upside down. For example, the reciprocal of \( \frac{3x^2}{24} \) is \( \frac{24}{3x^2} \).
• Multiply instead of divide. Replace the division with multiplication using the reciprocal. So, \( \frac{3x+6}{40} \div \frac{3x^2}{24} \) becomes \( \frac{3x+6}{40} \times \frac{24}{3x^2} \).

This method is very useful because multiplication is often a more straightforward operation than division, especially in algebraic expressions. This technique simplifies the process and reduces the potential for mistakes.

Factoring Algebraic Expressions

Factoring is a tool for breaking down expressions into simpler parts. It involves rewriting an expression as a product of its factors.Here's why factoring is important:

• Reduces complexity. By factoring, you're simplifying parts of the expression, making it easier to work with no matter the operation (addition, subtraction, multiplication, or division).
• Identifies common factors. Recognizing these helps in simplifying fractions by canceling out terms that appear both in the numerator and denominator.

In our problem, we see this with the numerator of the fraction \( \frac{3x+6}{40} \). It's factored as \( 3(x+2) \). This allows us to easily spot and cancel the \( 3 \) if it appears elsewhere in the fraction.Practicing factoring improves your ability to manipulate and solve algebraic expressions efficiently. It’s a foundational skill that simplifies many algebraic processes.