Problem 23 - Algebra for College Students

Question

Perform the following operations with real numbers. $$\frac{1}{2} \div\left(-\frac{1}{8}\right)$$

Step-by-Step Solution

Answer

The result is -4.

Step 1: Understand the Division Problem

The given problem is to divide \( \frac{1}{2} \) by \( -\frac{1}{8} \). This means we want to determine how many times \( -\frac{1}{8} \) fits into \( \frac{1}{2} \).

Step 2: Convert Division to Multiplication

Division by a fraction is equal to multiplication by its reciprocal. So, \( \frac{1}{2} \div -\frac{1}{8} \) can be re-written as \( \frac{1}{2} \times -8 \).

Step 3: Perform the Multiplication

Now, we multiply \( \frac{1}{2} \) by \(-8\). This can be calculated as \( \frac{1}{2} \times -8 = -\frac{8}{2} \).

Step 4: Simplify the Result

Simplify \( -\frac{8}{2} \) by performing the division: \( -\frac{8}{2} = -4 \). This is the result of our calculation.

Key Concepts

Division of Fractions

When you come across division involving fractions, the key is to realize that division can be turned into multiplication. The idea is based on multiplying by the reciprocal of the divisor. So, for the problem \( \frac{1}{2} \div \left(-\frac{1}{8}\right) \), you're essentially finding out how many times \( -\frac{1}{8} \) fits into \( \frac{1}{2} \).
To change the division into multiplication, flip the second fraction, \( -\frac{1}{8} \), to get \( -8 \). This is known as the reciprocal. Now, turn the division problem into a multiplication one: \( \frac{1}{2} \times -8 \). Remember: the reciprocal of a number is simply 1 divided by that number.

Multiplication and Division

Once you have expressed the division as multiplication by the reciprocal, the next step is actually doing the multiplication. We have \( \frac{1}{2} \times -8 \).
To multiply fractions or a fraction by a whole number:

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

In this case, \( \frac{1}{2} \times -8 \) is the same as multiplying 1 by -8 and then dividing by 2, which results in \( -\frac{8}{2} \).
This calculation brings us closer to the final answer. Multiplication and division often go hand in hand and knowing how to work with both helps streamline these processes.

Simplifying Fractions

The last step in solving problems involving fractions often involves simplification. Simplification is the process of reducing a fraction to its most straightforward form, where the numerator and denominator are the smallest whole numbers possible. For the solution \( -\frac{8}{2} \), we see that the fraction simplifies neatly.
To simplify fractions effectively:

• Look for common factors in the numerator and the denominator.
• Divide both by the greatest common divisor, if possible.

Here, dividing both the numerator and denominator by 2 simplifies \( -\frac{8}{2} \) to \( -4 \). Simplifying makes calculations more manageable and helps in getting a clearer final answer. It's a key skill in handling fractions and ensures your result is accurate and efficient.