Problem 57 - Elementary Algebra

Question

For exercises \(23-74\), evaluate. $$ -\frac{8}{15} \div \frac{3}{10} $$

Step-by-Step Solution

Answer

-\frac{16}{9}

Step 1: Understand the Division of Fractions

When dividing fractions, we multiply the first fraction by the reciprocal (or the inverse) of the second fraction.

Step 2: Find the Reciprocal of the Second Fraction

The reciprocal of \(\frac{3}{10}\) is \(\frac{10}{3}\).

Step 3: Set Up the Multiplication

Replace the division with multiplication by the reciprocal. So, \(-\frac{8}{15} \div \frac{3}{10}\) becomes \(-\frac{8}{15} \times \frac{10}{3}\).

Step 4: Multiply the Fractions

Multiply the numerators and the denominators: \(-\frac{8}{15} \times \frac{10}{3}\ = -\frac{8 \times 10}{15 \times 3}\ = -\frac{80}{45}\).

Step 5: Simplify the Fraction

Simplify \(-\frac{80}{45}\) by finding the greatest common divisor (GCD) of 80 and 45, which is 5. Divide both numerator and denominator by 5: \(-\frac{80 \div 5}{45 \div 5} = -\frac{16}{9}\).

Key Concepts

Reciprocal of a Fraction

When working with fractions, especially during division, it's crucial to understand the concept of the reciprocal. The reciprocal of a fraction is simply a flipped version of that fraction. For example, the reciprocal of \(\frac{3}{10}\) is \(\frac{10}{3}\).

To find a reciprocal, just swap the numerator (top number) and the denominator (bottom number).

• The reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).

In the context of our problem, we have to divide \(-\frac{8}{15} \) by \(\frac{3}{10}\). Instead of direct division, we multiply by the reciprocal. Thus, \(-\frac{8}{15} \div \frac{3}{10}\) becomes \(-\frac{8}{15} \times \frac{10}{3}\).

This forms the basis of the process, simplifying division into multiplication, which is usually easier to handle.

Multiplication of Fractions

Multiplication of fractions involves multiplying the numerators together and the denominators together. Given the transformed problem: \(-\frac{8}{15} \times \frac{10}{3}\), follow these steps:

• Multiply the numerators: \(-8 \times 10 = -80\)
• Multiply the denominators: \(\frac{15}{3} = 45\)

So, \(-\frac{8}{15} \times \frac{10}{3}= -\frac{80}{45}\).

It's helpful to remember that the rules for multiplying negative and positive numbers still apply. Here, a negative fraction times a positive fraction results in a negative fraction. This step is straightforward but vital for further simplification.

Simplification of Fractions

Simplifying fractions is done by dividing both the numerator and the denominator by their greatest common divisor (GCD). Let's simplify \(-\frac{80}{45}\):

• First, find the GCD of 80 and 45.
• The GCD of 80 and 45 is 5.
• Now, divide the numerator and the denominator by 5: \(-\frac{80 \div 5}{45 \div 5} = -\frac{16}{9}\)

This yields the simplified fraction \(-\frac{16}{9}\). Simplification can make fractions more manageable and easier to understand. Always check if your final answer can be simplified further.

In this exercise, we've transformed division into multiplication using reciprocals, performed the actual multiplication, and then simplified the result to its lowest terms. Understanding these steps makes complex fraction problems much easier to handle.