Problem 43 - 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. $$-32 \div 4$$

Step-by-Step Solution

Answer

The quotient of \(-32 \div 4\) is \(-8\).

Step 1: Part A: Convert the Division into Multiplication

To convert the division \(-32 \div 4\) into multiplication, replace the division by multiplication and replace the divisor 4 by its multiplicative inverse. The multiplicative inverse of any number \(x\) is \(1/x\), which can also be referred to as the reciprocal. Therefore, the multiplicative inverse of 4 is \(1/4\). So, \(-32 \div 4\) can be rewritten as \(-32 \times (1/4)\).

Step 2: Part B: Solve the Multiplication

To find the quotient, we perform the multiplication from Part A. Multiplying \(-32\) by \(1/4\), we get \(-8\). Therefore, the quotient of \(-32 \div 4\) is \(-8\).

Key Concepts

Division and Multiplication in Algebra

Division and multiplication in algebra often involve variables and constants and can sometimes be abstract for students. However, the principles are the same as with arithmetic. In multiplication, you are essentially adding a number to itself a certain number of times. Conversely, division is the process of splitting a number into equal parts.

For example, if you have the expression \(4x \div 2\), you can simplify this by remembering that dividing is the same as multiplying by the reciprocal. So, this expression becomes \(4x \times \frac{1}{2}\). Understanding this concept enhances solving algebraic expressions, making it less intimidating to work through more complex problems. Simplifying algebraic expressions by recognizing division as multiplication by a reciprocal is a powerful tool in algebra that can help students navigate through various mathematical challenges.

Reciprocal

The concept of a reciprocal is fundamental in mathematics, and particularly in algebra. In simple terms, the reciprocal of a number is what you multiply that number by to get 1. For any non-zero number \(x\), the reciprocal is \(1/x\). It's also called the multiplicative inverse because when you multiply a number by its reciprocal, you always get 1, hence the 'inverse' aspect.

Understanding reciprocals is crucial when dealing with division because division by a number is equivalent to multiplication by its reciprocal. For instance, dividing by 3 (\(\div 3\)) is the same as multiplying by \(1/3\) (\(\times \frac{1}{3}\)). This insight simplifies many calculations and allows for a smoother transition between forms of expressions, which can be especially useful when you're solving equations or simplifying expressions.

Solving Algebraic Expressions

Solving algebraic expressions can often be done by applying the rules of arithmetic operations and the properties of real numbers. The key to unlocking the solution is understanding the order of operations and the role of the multiplicative inverse in the division.

When you come across an algebraic expression that requires division, you can rewrite it to involve multiplication by the reciprocal. Doing so can sometimes make the expression easier to solve or simplify. For example, in the problem \( -32 \div 4 \), rewriting it as \( -32 \times \frac{1}{4} \) uses the concept of reciprocal to transform the division into a multiplication problem. This approach not only simplifies the calculation but also helps when dealing with more complex expressions that involve both variables and constants.