Problem 27 - Intermediate Algebra

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 Division with Fractions

Division by a fraction can be converted into multiplication by the reciprocal of the fraction. This means that instead of dividing by a fraction, you multiply by its reciprocal.

Step 2: Find the Reciprocal of the Divisor

The divisor in this problem is \(-\frac{1}{8}\). The reciprocal of \(-\frac{1}{8}\) is \(-8\) because the reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\), and we also change the sign.

Step 3: Convert Division to Multiplication

Now, the expression \(\frac{1}{2} \div \left(-\frac{1}{8}\right)\) changes to \(\frac{1}{2} \times (-8)\) by using the reciprocal of the divisor. This converts a division problem into a multiplication problem.

Step 4: Perform the Multiplication

Multiply \(\frac{1}{2}\) by \(-8\). To do this, multiply the numerator \(1\) by \(-8\) to get \(-8\), and keep the denominator as \(2\). So, the expression becomes \(\frac{-8}{2}\).

Step 5: Simplify the Result

Simplify \(-\frac{8}{2}\) by dividing the numerator and the denominator by their greatest common divisor, which is \(2\). This simplifies to \(-4\).

Key Concepts

Division of Fractions

Dividing by a fraction may initially seem tricky, but once you understand the rule, it becomes straightforward. When you divide by a fraction, you are essentially asking how many times that fraction fits into the dividend. Here’s a simple way to deal with the division of fractions:

• Find the reciprocal of the divisor (the fraction you are dividing by).
• Convert the division operation into multiplication by the reciprocal.

By doing this, the problem changes from a complex division problem to a simpler multiplication one. For example, in the problem \(rac{1}{2} \div \left(-\frac{1}{8}\right)\), instead of operating with division, you will multiply \(\frac{1}{2}\) by the reciprocal of \(-\frac{1}{8}\). This completes the transformation of the division into multiplication, making further calculations much easier.

Reciprocal of a Fraction

The reciprocal of a fraction is an essential concept in mathematics, especially in operations involving division. When finding the reciprocal, we simply flip the numerator and the denominator. For a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). This simple operation allows us to transform division into multiplication. However, be mindful of the sign. When dealing with negative fractions, you must maintain the negative sign even after flipping. In our earlier example with \(-\frac{1}{8}\), the reciprocal is \(-8\). This is because flipping the fraction also flips the position of the negative sign, but it remains intact. Once the reciprocal is identified, it is used to transform division into multiplication.

Multiplication by Reciprocal

In many mathematical operations, conversion of division to multiplication by the reciprocal simplifies the problem significantly. Once the reciprocal is found, performing the multiplication operation becomes a matter of traditional fraction multiplication. Here’s how to execute multiplication by the reciprocal step:

• Multiply the numerator of the first fraction by the whole reciprocal (here: integer form after turning the fraction's reciprocal).
• Retain the original denominator of the first fraction.

In our exercise, the calculation \(\frac{1}{2} \times (-8)\) gives \(-8\) for the numerator. The original denominator remains \(2\). Thus, the multiplication results in \(\frac{-8}{2}\). Lastly, simplify the fraction by dividing both the numerator and the denominator by their greatest common divider, leading to a more straightforward result—in this case, \(-4\). This process shows how division problems morph into simpler multiplication tasks, streamlining the math work significantly.