Problem 45 - 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. $$\frac{-60}{-5}$$

Step-by-Step Solution

Answer

The quotient of \(\frac{-60}{-5}\) is \(12\).

Step 1: Rewrite as Multiplication Using Multiplicative Inverse

To rewrite a division problem as a multiplication, you take the divisor and find its multiplicative inverse or reciprocal. The reciprocal of the number \(a\) is \(\frac{1}{a}\). In this case, the divisor is \(-5\) and its reciprocal is \(-\frac{1}{5}\). So, the division \(\frac{-60}{-5}\) can be rewritten as \(-60 \times -\frac{1}{5}\).

Step 2: Perform the Multiplication

Next, perform the multiplication. The multiplication of two negative numbers results in a positive number. So, \(-60 \times -\frac{1}{5}\) is equal to \(12\).

Step 3: Solution

The quotient of \(\frac{-60}{-5}\) or equivalently, \(-60 \times -\frac{1}{5}\), is \(12\).

Key Concepts

Division as Multiplication

Understanding how division can be thought of as multiplication is essential in algebra and simplifies solving division problems. This concept revolves around the use of the multiplicative inverse, also known as the reciprocal. To convert a division problem into a multiplication one, we multiply the dividend by the reciprocal of the divisor. For instance, dividing by a number is the same as multiplying by its reciprocal. So, if we have the division problem \( \frac{-60}{-5} \), we can convert it by finding the reciprocal of -5, which is \( -\frac{1}{5} \).

Thus, the division \( \frac{-60}{-5} \) becomes \( -60 \times -\frac{1}{5} \). Using this method reduces division problems to simpler multiplication problems, which are often easier to calculate, especially when dealing with negative numbers or fractions.

Reciprocal in Algebra

A reciprocal in algebra is simply the flipped version of a given non-zero number. For any non-zero number \( a \), its reciprocal is \( \frac{1}{a} \) and when multiplied together, the result is always 1. In the context of our exercise, the reciprocal of -5 is \( -\frac{1}{5} \).

Understanding reciprocals is crucial because they are used to simplify complex expressions and solve equations, particularly with division and fractions. In essence, finding the reciprocal is an important skill for restructuring equations and making them more approachable for further operations like multiplication or simplification.

Negative Numbers Multiplication

Multiplication involving negative numbers follows specific rules, which can sometimes be counterintuitive. The key rule to remember is that the product of two negative numbers is a positive number. This is critical when dealing with both division and multiplication operations involving negative numbers, as seen in our example.

When you multiply \( -60 \) by the reciprocal of \( -5 \), which is \( -\frac{1}{5} \), we have two negatives. Following the rule that a negative times a negative equals a positive, we compute \( -60 \times -\frac{1}{5} = 12 \). This understanding turns what may look like a daunting problem into a straightforward multiplication task.