Problem 10 - Elementary Algebra

Question

For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{4}{5}}{\frac{3}{10}} $$

Step-by-Step Solution

Answer

\(\frac{8}{3}\)

Step 1: Understand the Given Expression

You are given a compound fraction \(\frac{\frac{4}{5}}{\frac{3}{10}}\). The objective is to simplify this expression.

Step 2: Apply the Division Rule for Fractions

To divide by a fraction, you multiply by its reciprocal. Therefore, \(\frac{\frac{4}{5}}{\frac{3}{10}} = \frac{4}{5} \times \frac{10}{3}\).

Step 3: Perform the Multiplication

Multiply the numerators together and the denominators together: \(\frac{4 \times 10}{5 \times 3} = \frac{40}{15}\).

Step 4: Simplify the Resulting Fraction

To simplify \(\frac{40}{15}\), find the greatest common divisor (GCD) of 40 and 15, which is 5. Divide both the numerator and the denominator by 5: \(\frac{40 \rightarrow 40 \rightarrow 8}{15 \rightarrow 15 \rightarrow 3} = \frac{8}{3}\).

Key Concepts

Compound Fractions

A compound fraction, also known as a complex fraction, is a fraction where the numerator, the denominator, or both are themselves fractions. For example, in the given exercise, we have \(\frac{\frac{4}{5}}{\frac{3}{10}}\). Understanding compound fractions is crucial because they often appear in algebra and calculus. To simplify a compound fraction, you can follow these basic steps:

• Convert the division of fractions into a multiplication by the reciprocal
• Perform the multiplication
• Simplify the resulting fraction

Let's start with an example: \(\frac{\frac{4}{5}}{\frac{3}{10}}\). To simplify, first take the reciprocal of the denominator, \(\frac{3}{10}\), which becomes \(\frac{10}{3}\). Then multiply it by the numerator: \(\frac{4}{5} \times \frac{10}{3}\). Next, multiply the numerators and the denominators: \(\frac{4 \times 10}{5 \times 3} = \frac{40}{15}\). Finally, simplify the resulting fraction by finding a common factor, which leads us to the next core concept.

Reciprocal of a Fraction

The reciprocal of a fraction is simply a fraction flipped over, that is, the numerator becomes the denominator and the denominator becomes the numerator. Reciprocals are essential for dividing fractions.
For instance, in our exercise, we needed to divide by \(\frac{3}{10}\). So, we took its reciprocal, \(\frac{10}{3}\), and multiplied by it instead. Always remember:

• The reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\)
• Multiplying a fraction by its reciprocal will always equal 1, i.e., \(\frac{a}{b} \times \frac{b}{a} = 1\)

Using reciprocal operations facilitates the simplification or evaluation of compound fractions. After multiplying the numerators and the denominators, we often get a fraction that needs to be simplified to its lowest terms, requiring us to determine the greatest common divisor (GCD).

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) of two numbers is the largest positive integer that divides both without leaving a remainder. Finding the GCD is essential for simplifying fractions. It allows us to reduce fractions to their simplest form.
For example, in our solution to \(\frac{40}{15}\), we need to find the GCD to simplify it.

• The GCD of 40 and 15 is 5.
• We then divide both numerator and denominator by the GCD: \(\frac{40 \rightarrow 40\rightarrow 8}{15 \rightarrow 15 \rightarrow 3} = \frac{8}{3}\)

Here’s a quick guide on how to find the GCD:

1. List the factors of each number.
2. Identify the common factors.
3. Select the largest factor that both numbers share.

Simplifying fractions using the GCD ensures your final answer is in the simplest possible form, making it easier to understand and work with.