Problem 10 - Prealgebra

Question

Find the quotient in each case by replacing the divisor by its reciprocal and multiplying. $$\frac{3}{5} \div(-2)$$

Step-by-Step Solution

Answer

The quotient is \( \frac{-3}{10} \).

Step 1: Understand the Problem

The problem requires us to find the quotient of the fraction \( \frac{3}{5} \) when divided by \( -2 \). To do this, we will replace the division operation with multiplication by using the reciprocal of the divisor \( -2 \).

Step 2: Identify the Reciprocal of the Divisor

The divisor in the problem is \(-2\), whose reciprocal is \(-\frac{1}{2}\). This means instead of dividing by \(-2\), we'll multiply by \(-\frac{1}{2}\).

Step 3: Set Up the Multiplication

We will change the division problem \( \frac{3}{5} \div (-2) \) into a multiplication problem \( \frac{3}{5} \times -\frac{1}{2} \).

Step 4: Multiply the Fractions

To multiply \( \frac{3}{5} \times -\frac{1}{2} \), multiply the numerators and the denominators: \( 3 \times -1 = -3 \) and \( 5 \times 2 = 10 \). The result is \( \frac{-3}{10} \).

Step 5: Simplify the Result

The fraction \( \frac{-3}{10} \) is already in its simplest form, so this is the final result of the division.

Key Concepts

Reciprocal

A reciprocal is simply found when you "flip" a fraction. More precisely, a number's reciprocal is 1 divided by that number. If you have a whole number like

• 2, its reciprocal is \(\frac{1}{2}\).
• -2, its reciprocal becomes \(-\frac{1}{2}\).

For a fraction, such as \(\frac{3}{5}\), the reciprocal is \(\frac{5}{3}\). To solve division problems with fractions, always replace the division operation, by treating it as multiplication with the reciprocal of the divisor. This method converts a complex division task into an easier multiplication task, where we then just multiply as usual.

Multiplication of Fractions

Multiplying fractions is a straightforward process. Here’s how you do it:

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

For instance, if we multiply \(\frac{3}{5}\) by \(-\frac{1}{2}\), perform these steps:

• Multiply the numerators: \( 3 \times -1 = -3 \).
• Multiply the denominators: \( 5 \times 2 = 10 \).

After these calculations, you have the fraction \(\frac{-3}{10}\). Multiplying fractions is all about using these straightforward operations. Simplifying fractions often follows this step, which helps in making calculations easier and results simpler.

Simplifying Fractions

Simplifying fractions means reducing the fraction to its simplest form. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. To simplify, consider

• Finding the greatest common divisor (GCD) of the numerator and the denominator.
• Dividing both the top and bottom by the GCD.

In our example, \(\frac{-3}{10}\), the GCD is 1, so the fraction is already simplified. Simplifying ensures the result is the easiest to understand and work with. While not every fraction needs further simplification, it’s essential to check since fully simplified fractions can make further calculations more effortless.