Problem 43 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

The answer is \[ \frac{8}{3} \].

Step 1: Understand the problem

The problem requires dividing two fractions: \[ \frac{1}{3} \text{ and } \frac{1}{8} \]

Step 2: Recall division of fractions

To divide fractions, multiply the first fraction by the reciprocal of the second fraction. The reciprocal of \[ \frac{1}{8} \] is \[ \frac{8}{1} \].

Step 3: Perform multiplication

Multiply \[ \frac{1}{3} \] by \[ \frac{8}{1} \]: \[ \frac{1}{3} \times \frac{8}{1} = \frac{1 \times 8}{3 \times 1} = \frac{8}{3} \]

Step 4: Simplify the answer

The fraction \[ \frac{8}{3} \] is already in its simplest form.

Key Concepts

Fraction Division

Dividing fractions might sound complex, but it's actually straightforward once you get the hang of it. When faced with a problem like \(\frac{1}{3} \div \frac{1}{8}\), remember that division of fractions involves an extra step compared to whole numbers. Instead of directly dividing, you'll multiply the first fraction by the reciprocal of the second fraction.

Consider two key steps:

• First, rewrite the division problem using multiplication.
• Then, find the reciprocal of the second fraction and multiply the two fractions.

Following this method always works, allowing you to tackle any fraction division problem with confidence.

Reciprocal

Understanding the reciprocal is essential for dividing fractions. The reciprocal of a fraction is what you get when you flip the numerator (top number) and the denominator (bottom number). For example, the reciprocal of \(\frac{1}{8}\) is \(\frac{8}{1}\).

Using reciprocals converts a division problem into a multiplication problem. This is why, to divide by a fraction, you multiply by its reciprocal.
It's helpful to practice finding reciprocals of various fractions until it becomes second nature.

• The reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).
• Remember, the product of a fraction and its reciprocal is always 1. For instance, \(\frac{1}{8} \times \frac{8}{1} = 1\).

Mastering reciprocals is a key step in making fraction division easier.

Multiplying Fractions

Once you convert a division problem to multiplication using the reciprocal, the next step is to multiply the fractions. Multiplying fractions is more straightforward than it might seem.

Here's a quick guide:

• Multiply the numerators (top numbers) together to get the new numerator.
• Multiply the denominators (bottom numbers) together to get the new denominator.

For our problem, \(\frac{1}{3} \times \frac{8}{1}\), you multiply the numerators: 1 \times 8 = 8, and the denominators: 3 \times 1 = 3. The result is \(\frac{8}{3}\).
Lastly, always check if you can simplify the fraction by dividing the numerator and the denominator by their greatest common divisor. In this case, \(\frac{8}{3}\) is already in its simplest form. This simple process of multiplying fractions ensures you get the correct result every time.