Problem 8 - Prealgebra

Question

Find each quotient. Use an area model if necessary. $$\frac{4}{5} \div(-2)$$

Step-by-Step Solution

Answer

The quotient is \(-\frac{2}{5}\).

Step 1: Understand Division of Fractions or Numbers

We're asked to find the quotient of \(\frac{4}{5}\) divided by \(-2\). Dividing by a negative number means the result will be negative. The division of fractions involves multiplying by the reciprocal of the divisor.

Step 2: Convert "Division" to "Multiplication"

First, recognize that dividing by a number is the same as multiplying by its reciprocal. Here \(-2\) can be written as \(\frac{-2}{1}\). The reciprocal of \(\frac{-2}{1}\) is \(\frac{-1}{2}\). Thus, \(\frac{4}{5} \div (-2)\) becomes \(\frac{4}{5} \times \left(-\frac{1}{2}\right)\).

Step 3: Multiply the Fractions

To multiply the fractions, multiply the numerators together and the denominators together. The multiplication results in: \(\frac{4 \times (-1)}{5 \times 2} = \frac{-4}{10}\).

Step 4: Simplify the Fraction

The fraction \(\frac{-4}{10}\) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \(\frac{-4 \div 2}{10 \div 2} = \frac{-2}{5}\).

Key Concepts

Understanding the Reciprocal

The reciprocal of a number is simply flipping the numerator and the denominator. This is a key concept when it comes to dividing fractions. For example, the reciprocal of the whole number \(-2\) can be expressed as \(\frac{-2}{1}\), which is actually just \(-2\). The reciprocal of \(\frac{-2}{1}\) is \(\frac{-1}{2}\). When you multiply by the reciprocal, you are essentially flipping the operation from division to multiplication.
To divide fractions, replace the division sign with a multiplication sign and use the reciprocal of the divisor. This is the same as inverting the divisor and multiplying instead of dividing. Recognizing how to find and use a reciprocal is crucial for solving mathematical problems involving fraction division.

How to Multiply Fractions

Multiplying fractions is a straightforward process, and it involves two main steps: multiplying the numerators and multiplying the denominators. Rather than treating it as a series of complex operations, understand it as a simple matter of direct multiplication between two fractions.
For instance, consider multiplying \(\frac{4}{5}\) and \(\frac{-1}{2}\). This is performed as follows: multiply the numerators \(4 imes -1 = -4\) and the denominators \(5 imes 2 = 10\).
This operation results in the fraction \(\frac{-4}{10}\). Remember that when you multiply two numbers, the resulting fraction can sometimes appear more complicated, but it will often simplify to a simpler fraction.

Simplifying Fractions

Once you have multiplied the fractions, it's often necessary to simplify the result to its simplest form. Simplifying a fraction makes it easier to read and work with. You do this by dividing both the numerator and the denominator by their greatest common divisor (GCD).
For the fraction \(\frac{-4}{10}\), the GCD of 4 and 10 is 2. By dividing both the numerator and denominator by 2, you simplify the fraction to \(\frac{-2}{5}\).

• First, identify the GCD of the numerator and the denominator.
• Then, divide both parts of the fraction by this number.

This reduced fraction is often easier to understand and use in further calculations. Simplifying is a crucial step in many mathematical processes, allowing you to deal with clearer and more manageable numbers.