Problem 85 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ -5.2 \cdot 100 $$

Step-by-Step Solution

Answer

The result of the operation is -520.

Step 1: Understand the Components

We have the operation \(-5.2 \cdot 100\) which involves multiplying a negative decimal by a whole number.

Step 2: Apply Multiplication

To perform the operation, multiply the absolute values of the numbers. Calculate \(5.2 \times 100\): \(5.2 \times 100 = 520\).

Step 3: Apply the Negative Sign

Since one of the multiplicands is negative, the result will also be negative. Therefore, multiply the result by -1 to account for the sign: \(-1 \times 520 = -520\).

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero and are represented with a minus sign (-). They are found to the left of zero on a number line.- Negative numbers came into use to indicate quantities that are less than nothing, such as debts or temperatures below freezing.- An important property of negative numbers is that multiplying two negative numbers results in a positive product, whereas multiplying a negative and a positive number results in a negative product.- In our example, the number \(-5.2\) is negative, meaning it has a value less than zero.- When we multiply \(-5.2\) by any positive number, such as 100, the product will also be negative.- This happens because of the rule that when you multiply a negative number by a positive, the result is negative.Understanding these basics ensures that operations involving negatives don’t lead to errors.

Absolute Value

Absolute value refers to the distance of a number from zero on the number line, regardless of direction. It is always positive.- The absolute value is denoted by two vertical lines, like \(|x|\). For example, \(|-5.2| = 5.2\).- This concept is crucial in computations, especially when dealing with negative numbers.- In our problem, we first take the absolute value of the numbers involved.- Here, \(|-5.2| = 5.2\), which allows us to ignore the negative sign initially during computation.- By multiplying the absolute values, \((5.2 \times 100 = 520)\), we can simplify the multiplication.- We then return to the original operation to reintroduce the negative: \(-520\).Absolute value helps in focusing purely on numerical magnitude during calculations.

Whole Numbers Multiplication

Whole numbers are numbers without fractions or decimals. They include zero and all positive numbers (0, 1, 2, 3, etc.).-Multiplying whole numbers is one of the fundamental operations in math, straightforward because only the basic rules of arithmetic apply.- When multiplying a whole number, such as 100, by a decimal or any other number, we follow the standard method: multiply the digits as though the decimal were not there, and then adjust for the decimal's placement in the final product.- In this task, we need to multiply \(-5.2\) by 100. - By regarding 100 as a whole number and using absolute value, we efficiently compute \(5.2 \times 100 = 520\).- Multiplication of whole numbers underlies more complex operations, laying the groundwork for calculating with fractions and decimals.