Problem 46 - Introductory Algebra for College Students

Question

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

The quotient when -30 is divided by -5 is 6.

Step 1: Identify the Multiplicative Inverse

The multiplicative inverse of a number \(x\) is \(1/x\). In this case, the multiplicative inverse of \(-5\) is \(-1/5\).

Step 2: Rewrite the Division as Multiplication

The division \(-30/-5\) may be rewritten as a multiplication by the multiplicative inverse of \(-5\), which is \(-1/5\). So, the fraction is recreated as \(-30 * -1/5\).

Step 3: Perform the Multiplication

The multiplication \(-30 * -1/5\) gives the result \(+30/5\), which is 6.

Key Concepts

Multiplicative Inverse

Understanding the multiplicative inverse is essential when dealing with division and multiplication. The multiplicative inverse of a number is simply another number that, when multiplied by the original number, yields a product of 1. It's often referred to as the reciprocal. For any non-zero number \(x\), its multiplicative inverse is \(1/x\).

For example, the multiplicative inverse of \(2\) is \(1/2\), and when you multiply these two together, \(2 * 1/2 = 1\). When dealing with negative numbers, like \( -5\), its multiplicative inverse is \( -1/5\), and \( -5 * -1/5 = 1\). This concept turns division problems into multiplication problems, making them often simpler to solve, especially when working with variables and more complex algebraic expressions.

Changing Division to Multiplication

Division can be transformed into multiplication by using the concept of the multiplicative inverse. When you are asked to divide one number by another, such as \(\frac{-30}{-5}\), you can instead multiply by the inverse of the divisor (the number you are dividing by).

So, rewriting this division as multiplication would involve multiplying \( -30\) by the multiplicative inverse of \( -5\), which is \( -1/5\). It simplifies the division process and is especially useful when dealing with fractions, variables, and complex algebraic expressions. This approach is a valuable tool in algebra as it helps streamline quotient calculations.

Quotient Calculation

Calculating quotients is a straightforward process once you've transformed division into multiplication. After determining the multiplicative inverse, you multiply it by the number you are dividing. For the original problem \(\frac{-30}{-5}\), following the conversion to multiplication using the multiplicative inverse, the resulting operation is \( -30 * -1/5\).

Now, to determine the quotient, simply multiply these numbers. Since a negative times a negative yields a positive result, \( -30 * -1/5 = 30/5 = 6\). This final answer is the quotient, which is the result you get when you divide one number by another.

Algebraic Operations with Negative Numbers

Algebraic operations with negative numbers follow specific rules that ensure consistent results. The basic operations—addition, subtraction, multiplication, and division—have their own rules when it comes to negative numbers.

For multiplication and division, the rules are as follows: multiplying or dividing two negative numbers gives a positive result, while doing so with one negative and one positive number gives a negative result. For instance, in the multiplication \( -30 * -1/5\), both numbers are negative, hence their product is positive, resulting in 6. Understanding these rules is crucial for correctly carrying out operations and solving algebraic expressions that involve negative numbers.