Problem 5 - Prealgebra

Question

Find the quotient in each case by replacing the divisor by its reciprocal and multiplying. $$6 \div\left(-\frac{2}{3}\right)$$

Step-by-Step Solution

Answer

The quotient is \(-9\).

Step 1: Identify the Division Expression

We are given the division expression \( 6 \div \left(-\frac{2}{3}\right) \). We need to find the quotient by changing the division operation into multiplication.

Step 2: Find the Reciprocal of the Divisor

The divisor here is \(-\frac{2}{3}\). The reciprocal of \(-\frac{2}{3}\) is \(-\frac{3}{2}\).

Step 3: Change Division to Multiplication

Now replace the division operation with multiplication using the reciprocal of the divisor. Therefore, \(6 \div \left(-\frac{2}{3}\right) \) becomes \( 6 \times \left(-\frac{3}{2}\right) \).

Step 4: Multiply the Numbers

To find the product, multiply 6 by \(-\frac{3}{2}\). This can be calculated as \( 6 \times \left(-\frac{3}{2}\right) = 6 \times -1.5 = -9 \).

Step 5: Conclusion

The quotient of the original expression \( 6 \div \left(-\frac{2}{3}\right) \) is \(-9\).

Key Concepts

Understanding Reciprocals

In the world of fractions, the reciprocal plays an essential role, especially when dividing fractions. A reciprocal is simply what you multiply a number by to get 1. For any fraction, its reciprocal is obtained by swapping its numerator and denominator. For example, the reciprocal of \(-\frac{2}{3}\) is \(-\frac{3}{2}\). Notice that the reciprocal keeps the same sign, be it positive or negative. Thus, if you start with a negative fraction, its reciprocal is also negative.

• To find the reciprocal of a fraction, simply interchange its top (numerator) and bottom (denominator) parts.
• The product of a number and its reciprocal is always 1.
• Reciprocals are crucial for converting division into multiplication.

Multiplying Fractions Made Easy

Multiplying fractions might initially seem challenging, but it's straightforward with a little practice. To multiply two fractions, you only need to multiply the numerators (the numbers on top of the fractions) and then multiply the denominators (the numbers on the bottom). For example, consider multiplying \(6\) by \(-\frac{3}{2}\). First, treat \(6\) as \(\frac{6}{1}\). Then multiply: \(6 \times -3 = -18\) and \(1 \times 2 = 2\). Thus, \(6 \times \left(-\frac{3}{2}\right) = \frac{-18}{2}\), which simplifies to \(-9\).

• Always multiply numerators with numerators, and denominators with denominators.
• Before multiplying, try to simplify fractions if possible, which makes your calculation easier.
• Remember, the final answer should always be simplified to its simplest form.

The Role of Negative Numbers in Division

In mathematics, dealing with negative numbers can be a bit tricky, especially with multiplication and division. Negative numbers have special rules that help us maintain balance and derive correct results. When multiplying or dividing:

• A positive number by a negative number always results in a negative outcome.
• A negative number multiplied or divided by another negative results in a positive outcome.
• When dealing with division, like in the original exercise \(6 \div \left(-\frac{2}{3}\right)\), changing the operation to multiplication by the negative reciprocal transforms the division into multiplication of a positive and a negative. The resultant is negative, which matches our expectations.

These rules ensure we handle operations correctly when negative numbers are involved.