Problem 46 - Basic College Mathematics

Question

Simplify. $$ \frac{\frac{1}{2}}{\frac{1}{8}} $$

Step-by-Step Solution

Answer

4

Step 1: Understand the Problem

We need to simplify the fraction \(\frac{\frac{1}{2}}{\frac{1}{8}} \). This requires dividing one fraction by another.

Step 2: Apply the Division Rule for Fractions

To divide by a fraction, multiply by its reciprocal. The reciprocal of \( \frac{1}{8} \) is \( 8 \). So, our problem becomes \( \frac{1}{2} \times 8 \).

Step 3: Perform the Multiplication

Now, multiply \( \frac{1}{2} \) and \( 8 \): \(\frac{1}{2} \times 8 = 4 \).

Step 4: Simplify the Result

The result of \( \frac{\frac{1}{2}}{\frac{1}{8}} \) simplifies to \( 4 \).

Key Concepts

Dividing Fractions

Dividing fractions might seem complicated at first, but it's actually quite straightforward once you learn the steps. The key idea is to multiply by the reciprocal of the divisor. To divide fractions, follow these simple steps:

1. Identify the two fractions involved. For example, in the exercise, the fractions are \(\frac{1}{2}\) and \(\frac{1}{8}\).
2. Take the reciprocal of the second fraction (the divisor). The reciprocal of a fraction is achieved by swapping its numerator and denominator. Thus, the reciprocal of \(\frac{1}{8}\) is \(8\).
3. Multiply the first fraction (the dividend) by the reciprocal of the second fraction. In our case, we multiply \(\frac{1}{2}\) by \(8\).
4. Simplify the resulting fraction if possible. Multiplying \(\frac{1}{2}\) by \(8\) gives us \(4\), which is already in its simplest form.

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is crucial for dividing fractions. The reciprocal of a fraction is what we get when we exchange its numerator (top part) and its denominator (bottom part).

For example:

• The reciprocal of \( \frac{1}{2} \) is \( 2 \) because \( \frac{1}{2} \) becomes \( \frac{2}{1} \), which simplifies to \(2\).
• The reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \).
  
• The reciprocal of a whole number like \( 5 \) is \( \frac{1}{5} \).
  

Knowing how to find reciprocals makes it much easier to divide fractions, as seen in the exercise where the reciprocal of \( \frac{1}{8} \) is \( 8 \). When dividing by a fraction, you effectively multiply by its reciprocal.

Multiplication of Fractions

Multiplying fractions is another important concept that is used when dividing fractions. Once you have the reciprocal, you need to multiply the fractions.

Here’s how to multiply fractions:

• Multiply the numerators (top numbers) together.
• Multiply the denominators (bottom numbers) together.

Let's apply this to our problem. When multiplying \( \frac{1}{2} \) by \( 8 \), we consider \( 8 \) as \( \frac{8}{1} \). Now we do the multiplication:

\[\frac{1}{2} \times \frac{8}{1} = \frac{1 \times 8}{2 \times 1} = \frac{8}{2} = 4\]
This multiplication gives us the simplified result of \( 4 \). Remember, the operations with fractions always follow these rules, so mastering them will make all fraction problems easier to tackle.