Problem 30 - Prealgebra

Question

Find each quotient. Use an area model if necessary. $$\frac{2}{9} \div\left(-\frac{2}{9}\right)$$

Step-by-Step Solution

Answer

The quotient is -1.

Step 1: Understand the Division of Fractions

To divide fractions, we multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is the inverse of the fraction; in other words, you swap the numerator and denominator.

Step 2: Identify the Fractions and Their Reciprocals

The expression given is \( \frac{2}{9} \div \left(-\frac{2}{9}\right) \). The reciprocal of \(-\frac{2}{9}\) is \(-\frac{9}{2}\).

Step 3: Multiply the Fractions

To find the quotient of \( \frac{2}{9} \div \left(-\frac{2}{9}\right) \), multiply \( \frac{2}{9} \) by \( -\frac{9}{2} \). Perform the multiplication: \[\frac{2}{9} \times \left(-\frac{9}{2}\right) = -\frac{18}{18} = -1.\]

Step 4: Simplifying the Product

The multiplication gives \(-\frac{18}{18}\). Simplify \(-\frac{18}{18}\): \[-\frac{18}{18} = -1.\]Thus, the quotient is \(-1\).

Key Concepts

Reciprocal

In mathematics, the concept of a reciprocal is a foundational tool for dividing fractions. The reciprocal of a number is simply another number that, when multiplied with the original number, yields 1. To find the reciprocal of a fraction, you swap its numerator and denominator.

For instance, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \). The process is straightforward:

• Identify the numerator (top number) and the denominator (bottom number) of the fraction.
• Swap their positions.

This way, multiplying a fraction by its reciprocal results in 1, as the product of the numerator and denominator equals the product of the denominator and numerator.

In the context of dividing fractions, like in the exercise \( \frac{2}{9} \div \left(-\frac{2}{9}\right) \), you take the reciprocal of \(-\frac{2}{9}\), which is \(-\frac{9}{2}\). This step shifts the division into multiplication, simplifying the process.

Integer Division

Understanding integer division is crucial when dealing with the division of fractions, especially when the fractions involve integers. An integer can be a positive or negative whole number. In division, when dividing fractions, the reciprocal conversion turns the division into multiplication.

Consider a situation with integers: if you need to divide two fractions like \( \frac{4}{5} \div \left(\frac{2}{3}\right) \), you would change the division into multiplication by the reciprocal of the second fraction (\( \frac{3}{2} \)). You end up performing multiplication between fractions that may involve integers in the operations:

• Determine the fractions to be divided.
• Find the reciprocal of the divisor (fraction that divides into another).
• Convert the division into a multiplication problem.
• Multiply using straightforward fraction multiplication rules.

Ensure the final answer is properly simplified. Although the integers in fractions make the process seem complex, applying these steps simplifies division vastly, even when negatives or improper fractions are involved.

Negative Numbers

Handling negative numbers in fraction division is an important skill. Negative numbers are numbers less than zero, and they impact the sign of the quotient in division.

When dividing fractions, if one of the fractions is negative, the entire division result will also be negative. Here’s a simple guideline:

• One negative fraction leads to a negative quotient.
• If both fractions are negative, the negatives cancel, making the quotient positive.
• Use the reciprocal rule to handle the division operation.

Returning to our example: when dividing \( \frac{2}{9} \) by \(-\frac{2}{9} \), we switch from division to multiplication by taking the reciprocal \(-\frac{9}{2}\). With the multiplication \( \frac{2}{9} \times -\frac{9}{2} \), the product is \(-\frac{18}{18} = -1\). Thus, a negative number will influence the product's sign as shown here. Understanding how negatives interact within dividing fractions ensures that you get the sign of the answer right every time.