Problem 57 - Prealgebra

Question

Use the rule for order of operations along with the rules for addition, subtraction, and multiplication to simplify each of the following expressions. What number do you multiply by \(-3\) to get \(12 ?\)

Step-by-Step Solution

Answer

The number is -4.

Step 1: Identify the Problem

We are given that \(-3\times\text{something} = 12 \)and need to find the 'something'.

Step 2: Rearrange the Equation

Rearrange the equation to solve for the unknown number. Start with: \(x \times (-3) = 12\).

Step 3: Isolate the Unknown

Divide both sides of the equation by \(-3\) to isolate \(x\). \(x = \frac{12}{-3}\).

Step 4: Calculate the Division

Perform the division to find \(x\). \(x = -4 \).

Key Concepts

Understanding Multiplication

Multiplication is one of the basic operations in mathematics and involves repeated addition of numbers. When you multiply a number, it simply means adding that number to itself a specified number of times. For example, if you multiply 4 by 3, you are adding 4 three times (4 + 4 + 4 = 12). In our original problem, we have the expression

• \(-3 \times ? = 12\).

Here, multiplication is used with a negative number, but the underlying concept of repeated addition remains the same. The ? or unknown can be thought of as the number that needs to be multiplied with \(-3\) to yield \(12\). The goal is to determine the missing factor in this multiplication equation.

Mastering Division

Division is essentially the opposite of multiplication. If multiplication involves adding something repeatedly, division is about splitting something into equal parts. When faced with an equation like

• \(x \times (-3) = 12\),

division helps us find the unknown factor \(x\). In this context, to "undo" the multiplication by \(-3\), we divide the other side of the equation by \(-3\). This action provides a systematic way to isolate the variable on one side of the equation:

• \(x = \frac{12}{-3}\).

The result of this division is \(-4\), meaning when \(-4\) is multiplied by \(-3\), we indeed get \(12\). Always remember, division breaks things down, just as multiplication builds things up.

Negative Numbers and Their Properties

Negative numbers, which are numbers less than zero, play a crucial role in mathematics. When dealing with negative numbers, it is essential to understand how they interact with other operations, like multiplication and division. For instance, the multiplication or division of two negative numbers yields a positive result.

• Multiplying a positive number by a negative number reverses the sign of the result.

In our exercise, we had a negative number multiplying an unknown to produce a positive number. This scenario suggested that the unknown itself must be negative, aligning with the principle that a negative times a negative gives a positive. Thus,

• \(-3 \times (-4) = 12\).

Grasping the properties of negative numbers will help you analyze and solve equations involving them more effectively.