Problem 80 - Beginning and Intermediate Algebra

Question

Divide. $$ \frac{4}{5} \div\left(-\frac{1}{2}\right) $$

Step-by-Step Solution

Answer

Divide by multiplying with the reciprocal: \(-\frac{8}{5}\).

Step 1: Understand the Division Problem

The problem is asking you to divide \( \frac{4}{5} \) by \( -\frac{1}{2} \). This is equivalent to finding how many times \( -\frac{1}{2} \) fits into \( \frac{4}{5} \). Instead of direct division, we will multiply by the reciprocal.

Step 2: Identify the Reciprocal

To solve a division problem involving fractions, we take the reciprocal of the divisor. The divisor in this case is \(-\frac{1}{2}\), so the reciprocal is \(-2\).

Step 3: Multiply Instead of Divide

Replace the division operation with multiplication by the reciprocal: \( \frac{4}{5} \times (-2) \).

Step 4: Perform the Multiplication

Multiply the numerators: \(4 \times (-2) = -8\). Multiply the denominators: \(5 \times 1 = 5\).

Step 5: Simplify the Result

The fraction obtained is \(\frac{-8}{5}\). This fraction is already in its simplest form.

Key Concepts

Reciprocal

When tackling fraction division, reciprocal is a term you might frequently encounter. It's the key to transforming a division problem into an easier multiplication problem.
In simple terms, the reciprocal of a fraction is when you flip the numerator and the denominator. This is done to switch a division operation into a multiplication. For example, if your fraction is \( \frac{1}{2} \), its reciprocal is \( 2 \), because you flip the positions of 1 and 2.
When dealing with a negative fraction, like \( -\frac{1}{2} \), its reciprocal is \( -2 \). The negative sign stays the same; only the positions of the numerator and denominator trade places.
To solve a fraction division problem, like \( \frac{4}{5} \div -\frac{1}{2} \), you start by identifying the reciprocal of the divisor, which is the fraction following the division sign. In this case, you take \( -\frac{1}{2} \) and find its reciprocal, \( -2 \). This small flip is essential because it allows us to switch from dividing to multiplying.

Multiplying Fractions

Once you have the reciprocal, multiplication is the next step in solving fraction division. You will use this method to arrive at your final answer.
In multiplying fractions, you do not need to find a common denominator. You simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. Let's see how this works with our exercise problem:

• Starting with \( \frac{4}{5} \) and multiplying it by its divisor's reciprocal, \( -2 \), you perform the operation: \( \frac{4}{5} \times -2 \).
• Multiply the numerators: \( 4 \times (-2) = -8 \).
• Multiply the denominators: \( 5 \times 1 = 5 \).

Therefore, the result of the multiplication is \( \frac{-8}{5} \). Multiplying fractions in this way is straightforward once you know the steps, and it's necessary to convert division problems into multiplication format.

Simplifying Fractions

Sometimes after performing operations with fractions, the fraction you end up with can be further simplified. Simplifying a fraction means reducing it to its simplest form, where the numerator and denominator share no common factors except for 1.
In our division example, we have arrived at \( \frac{-8}{5} \) after multiplying. Since this fraction is in proper form (numerator has a smaller absolute value than the denominator) and there's no number that will divide evenly into both 8 and 5 aside from 1, it's already in its simplest form.
To simplify fractions, you need to find the greatest common divisor (GCD) of both the numerator and the denominator. If this number is greater than 1, you can divide both by the GCD to reduce the fraction.
If you encounter a larger fraction, applying the GCD will quickly get you to the simplest form, ensuring your solution is both correct and neatly presented.