Problem 70 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(47-76\), perform the indicated division or state that the expression is undefined. $$-\frac{1}{2} \div\left(-\frac{7}{9}\right)$$

Step-by-Step Solution

Answer

The result of the division \(-\frac{1}{2} \div -\frac{7}{9}\) is \(\frac{9}{14}\).

Step 1: Identify the Numerator and Denominator

In the expression \(- \frac{1}{2} \div -( \frac{7}{9})\), the first fraction is \(- \frac{1}{2}\) and the second is \(-\frac{7}{9}\). When you divide fractions, you are asked to divide the first fraction (the numerator) by the second (the denominator)

Step 2: Apply the Rule of Signs

Since there are two negative signs for two fractions, they cancel each other out. Negative divided by negative yields a positive result. This modifies our expression into: \(\frac{1}{2} \div \frac{7}{9}\)

Step 3: Change the Division to Multiplication

Remember that division by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator. The reciprocal of \(\frac{7}{9}\) is \(\frac{9}{7}\). This changes our expression into: \(\frac{1}{2} \times \frac{9}{7}\)

Step 4: Multiply the Fractions

To multiply fractions, simply multiply the numerators together to get the new numerator, and the denominators together to get the new denominator. This gives us: \(\frac{1 \times 9}{2 \times 7} = \frac{9}{14}\)

Key Concepts

Reciprocal

When dividing fractions, understanding reciprocals is crucial. A reciprocal of a fraction is what you use to "flip" it upside down. In more formal terms, the reciprocal of a fraction is formed by swapping its numerator (the top number) and its denominator (the bottom number). This is important because dividing by a fraction is equivalent to multiplying by its reciprocal.

For example, the reciprocal of \( \frac{7}{9} \) is \( \frac{9}{7} \). Notice how the 7 and 9 switch places. This step is essential because when dividing fractions, you change the operation from division to multiplication by the reciprocal.

Always remember:

• The reciprocal of a fraction multiplies it into a whole number, if multiplied properly to the original.
• Reciprocals help convert division problems into simpler multiplication equations.

Mastering reciprocals makes fraction operations more intuitive and manageable.

Rule of Signs

The rule of signs is a handy guideline when working with negative and positive numbers, and it applies to multiplying and dividing as well. Essentially, we're interested in knowing whether the result will be a positive or negative number.

Here are some key points to remember:

• A negative divided by a negative results in a positive.
• A positive divided by a positive results in a positive.
• A negative divided by a positive, or a positive divided by a negative, results in a negative.

In the exercise, we began with \(- \frac{1}{2} \div -\frac{7}{9}\). Since we were dividing two negative numbers, our resulting quotient was positive \( \frac{1}{2} \div \frac{7}{9}\). This rule simplifies division problems, allowing you to focus on magnitudes without worrying about sign multiplicity.

Multiplying Fractions

Once you're past the rule of signs and have incorporated reciprocals, you need to multiply fractions. Multiplying is straightforward and involves two main steps: multiply the numerators and then the denominators. This is efficient and direct.

Let's break it down:
When multiplying two fractions like \( \frac{1}{2} \times \frac{9}{7} \), you multiply the numerators (1 and 9) together to get the new numerator. Then, you multiply the denominators (2 and 7) to get the new denominator. Therefore, the final result is \( \frac{1 \times 9}{2 \times 7} = \frac{9}{14} \).

• You don't need to worry about finding a common denominator, unlike adding or subtracting fractions.
• If possible, simplify your result by finding the greatest common divisor (GCD), although in our case, \( \frac{9}{14} \) is already simplified.

By practicing these steps, you can become quick and efficient at fraction multiplication.