Problem 134 - Introductory Algebra for College Students

Question

A multiplication is expressed as a repeated addition. Find this sum, indicated by a question mark. $$4(-3)=(-3)+(-3)+(-3)+(-3)=?$$

Step-by-Step Solution

Answer

-12

Step 1: Interpret the Multiplication Expression

The term \(4(-3)\) represents the multiplication of the number 4 and -3. Multiplication can be regarded as repeated addition. This implies that -3 is added four times.

Step 2: Apply Repeated Addition

Make the addition of -3, four times to replicate the multiplication of 4 and -3. This is expressed as \((-3) + (-3) + (-3) + (-3)\)

Step 3: Calculate the Sum

Add together the four instances of -3 to give the result. So, \((-3) + (-3) + (-3) + (-3) = -12\).

Key Concepts

Repeated Addition

Multiplication is essentially a way to simplify repeated addition. Instead of adding the same number several times, multiplication combines all those additions into a single operation. For instance, when we multiply 4 by -3, it means that we add -3 together, four times.
This can be expressed as

• \(-3 + (-3) + (-3) + (-3)\).

Each time we add, we reach a new sum. Through multiplication, this repetitive addition gives a quick and convenient result. So rather than adding manually each time, multiplication streamlines the process into one calculation.

Negative Numbers

Understanding negative numbers is crucial in arithmetic operations, especially in multiplication. Negative numbers are simply numbers with a value less than zero. They are often used to represent things like losses or decreases.
For example, in our exercise: 4 times -3, where -3 is a negative number, repeated adding of -3 results in each individual result being less than zero.
Mathematically, when you add negative numbers multiple times, you decrease the original number further. Hence,

• with repeated addition of -3 four times, the result is: \((-3) + (-3) + (-3) + (-3)\),

which simplifies to -12. This demonstrates how negative numbers influence calculations by driving the result downwards into the negative zone.

Arithmetic Operations

Arithmetic operations form the basis of mathematics, including addition, subtraction, multiplication, and division. These operations help us in solving everyday problems and calculations.
In our problem, multiplication is used as a primary operation, expressed through repeated addition.
Multiplication of a positive and a negative number always results in a negative number. Here, multiplying 4 (a positive) by -3 (a negative) yields -12.

• The consistent pattern: positive times negative equals negative, is used across arithmetic and is evident in our calculations.
• Understanding this relationship in operations helps solve equations more effectively by structuring the operation correctly.

Mastering these operations enables solving complex problems by breaking them down into simpler steps, just as we demonstrated in the given exercise.