Problem 45 - Elementary Algebra

Question

For exercises \(23-74\), evaluate. $$ \frac{1}{8} \div \frac{1}{3} $$

Step-by-Step Solution

Answer

The result of \( \frac{1}{8} \text{ divided by }\ \frac{1}{3} \) is \( \frac{3}{8} \).

Step 1 - Understanding the problem

The problem requires dividing two fractions: \( \frac{1}{8} \text{ and }\ \frac{1}{3} \).

Step 2 - Invert the second fraction

When dividing fractions, the second fraction (the divisor) should be inverted. The reciprocal of \( \frac{1}{3} \) is \( \frac{3}{1} \).

Step 3 - Multiply the fractions

Change the division to multiplication and multiply the first fraction by the reciprocal of the second: \( \frac{1}{8} \times \frac{3}{1} \).

Step 4 - Perform the multiplication

Multiply the numerators and the denominators: \( \frac{1 \times 3}{8 \times 1} = \frac{3}{8} \).

Step 5 - Simplify the fraction

In this case, the fraction \( \frac{3}{8} \) is already in its simplest form.

Key Concepts

Dividing fractions

Dividing fractions might seem tricky at first, but it's quite simple once you understand the process. The key rule to remember is that dividing by a fraction is the same as multiplying by its reciprocal. To divide fractions, follow these steps:

• Identify the fractions you need to divide.
• Find the reciprocal of the second fraction (the divisor).
• Multiply the first fraction by this reciprocal.
• Simplify the resulting fraction if necessary.

In the given exercise, we start with dividing \(\frac{1}{8} \div \frac{1}{3}\).

Reciprocal of a fraction

The reciprocal of a fraction is simply switching the numerator (top number) and the denominator (bottom number). This step is crucial in fraction division. For example:

• The reciprocal of \(\frac{1}{3}\) is \(\frac{3}{1}\).
• If you have \(\frac{5}{4}\), its reciprocal is \(\frac{4}{5}\).

To divide fractions, we replace the division with multiplication by the reciprocal. In our problem, \(\frac{1}{8} \div \frac{1}{3}\) becomes \(\frac{1}{8} \times \frac{3}{1}\).

Multiplication of fractions

Once you've found the reciprocal, the next step is straightforward multiplication.

• Multiply the numerators together.
• Multiply the denominators together.

This means for our exercise, we compute: \(\frac{1}{8} \times \frac{3}{1} = \frac{1 \times 3}{8 \times 1} \). This results in \(\frac{3}{8}\), which is the product of the multiplication.

Simplifying fractions

Simplifying fractions means reducing them to their simplest form. This isn’t always necessary if the resulting fraction is already simplified. To simplify:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by this GCD.

In our example, \(\frac{3}{8}\) is already in its simplest form because there are no common factors between 3 and 8 other than 1. So, \(\frac{3}{8}\) is the final answer.