Problem 46 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(43-46\) a. Rewrite the division as multiplication involving a multiplicative inverse. b. Use the multiplication from part ( \(a\) ) to find the given quotient. $$\frac{-30}{-5}$$

Step-by-Step Solution

Answer

Therefore, the quotient of \(\frac{-30}{-5}\) is 150.

Step 1: Understand the multiplicative inverse

A multiplicative inverse of a number is one which, when the number is multiplied by it results in the number 1. So for any number \(a\), its multiplicative inverse is \(\frac{1}{a}\). Now if \(a\) is a divisor, and \(-b\) is the dividend, the division \(\frac{-b}{a}\) can be rewritten as multiplication by changing the division to multiplication and replacing the divisor with its multiplicative inverse: \(-b \times \frac{1}{a}\)

Step 2: Rewrite the given division as multiplication

Here, you have the division \(\frac{-30}{-5}\). Use the converter in the previous step to rewrite it. It will be \(-30 \times \frac{1}{-5}\)

Step 3: Execute the multiplication to find the quotient

In the previous step, the division was rewritten as multiplication: \(-30 \times \frac{1}{-5}\). Now perform this multiplication to find the quotient. Two negative numbers result in a positive number when they're multiplied together, so it would simplify to \(30 \times 5\) which equals 150

Key Concepts

Division as Multiplication

Understanding division as multiplication involves using the idea of the multiplicative inverse. When you divide, you're essentially finding how many times one number fits into another. However, by turning this process into multiplication, you can sometimes simplify your calculations.

The key is to use a multiplicative inverse. The multiplicative inverse of any number \( a \) is \( \frac{1}{a} \). So, instead of dividing by a number, you multiply by its inverse. This means that a division problem like \( \frac{-b}{a} \) becomes \( -b \times \frac{1}{a} \).

For example, consider the division \( \frac{-30}{-5} \). Instead of dividing, this can be rewritten as \(-30 \times \frac{1}{-5}\). This switch from division to multiplication makes it often easier to handle, especially when dealing with more complex numbers or fractions.

Quotient Calculation

Calculating the quotient involves solving the multiplication problem created from the previous transformation. Let's revisit our example of \( \frac{-30}{-5} \), which was rewritten as \(-30 \times \frac{1}{-5}\).

To find the quotient, you multiply the dividend by the inverse of the divisor:

• First, identify that \(-30 \times \frac{1}{-5}\) involves multiplying two negative numbers.
• When two negative numbers are multiplied, the result is positive. Therefore, \(-30 \times \frac{1}{-5}\) becomes \(30 \times 5\).

The simple multiplication \(30 \times 5\) results in the quotient 150. Through this method of turning division into multiplication, calculating the quotient becomes straightforward and clear.

Negative Number Division

Handling the division of negative numbers can seem confusing. However, it's straightforward once we use the multiplicative inverse method from before.

When dividing negative numbers like \(-30\) by \(-5\), remember these essential points:

• Two negatives multiplied together give a positive result. This is crucial in ensuring we end up with the correct positive quotient.
• Use the inverse: Transform the division into multiplication with an inverse, as shown with \(-30 \times \frac{1}{-5}\).

As negative numbers interact, their signs dictate the direction, but the numbers themselves multiply as usual.

The important part is understanding that despite the presence of negative signs, the operational adjustments (like using the multiplicative inverse) help derive the positive final quotient, which in our example is 150. Remember these steps and negative divisions become just as manageable as any other numerical operation.