Problem 90 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ \frac{440}{-20} $$

Step-by-Step Solution

Answer

The result is -22.

Step 1: Identify the Operation

First, identify the mathematical operation required. This expression is a division problem that divides 440 by -20.

Step 2: Determine the Sign of the Result

Since we are dividing two numbers with different signs (440 is positive and -20 is negative), the result of the division will be negative.

Step 3: Perform the Division

Perform the actual division ignoring the signs for a moment. Divide 440 by 20: 440 ÷ 20 = 22.

Step 4: Apply the Sign to the Result

Now apply the negative sign from Step 2 to the result of the division. Hence, the result is -22.

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign (-). They are commonly used to represent opposite quantities, such as debts or temperatures below zero. Understanding negative numbers is essential because they are a fundamental part of mathematics.

Negative numbers appear everywhere in real life, from financial calculations to scientific measurements. Here are some key points about negative numbers:

• Negative numbers are to the left of zero on a number line.
• When you add a negative number, you actually subtract its value.
• When you subtract a negative number, it is the same as adding its positive equivalent.
• Multiplication and division rules with negative numbers are tied to working with different signs, which will be discussed in detail in the section about division with negative numbers.

Grasping the concept of negative numbers is crucial for solving equations and understanding various mathematical concepts. With practice, handling negative numbers becomes straightforward.

Mathematical Operations

Mathematical operations are processes that we use to find a result from numbers or expressions. The basic operations include addition, subtraction, multiplication, and division. Each operation follows specific rules that help us solve mathematical problems efficiently.

Here's a brief overview of these operations:

• Addition: Combines two or more numbers to find their total.
• Subtraction: Finds the difference between numbers and involves 'taking away'.
• Multiplication: Involves repeated addition of a number. For example, 3 times 4 is the same as adding 4 three times: 4 + 4 + 4.
• Division: Involves splitting a number into equal parts.

Understanding mathematical operations is vital for solving more complex problems involving algebraic expressions and equations. Each operation interacts with numbers uniquely, including whether they are positive or negative, which influences the outcome.

Division with Negative Numbers

Division is a mathematical operation where we split a number into equal parts. When dealing with negative numbers, the rules of division take on a particular significance.

The sign of the result in division depends on the signs of the numbers involved:

• If both numbers are positive or both are negative, the result is positive.
• If one number is negative and the other is positive, the result is negative.

This can be summarized by the rules of multiplication sign simplification:

• Negative divided by positive results in a negative.
• Positive divided by negative also results in a negative.
• Negative divided by negative gives a positive.

Let's apply these rules to the problem at hand, \( \frac{440}{-20} \):

1. Calculate as if both numbers were positive, i.e., \( \frac{440}{20} \), which equals 22.
2. Since one number is negative, apply a negative sign to the result, making it \(-22\).

Understanding how division with negative numbers works is crucial for solving not just simple arithmetic problems, but also complex algebraic equations. With these foundational rules, you can tackle a wide variety of division problems effectively.