Problem 75 - Elementary and Intermediate Algebra

Question

Add. $$ 19.2+(-41.3) $$

Step-by-Step Solution

Answer

-22.1

Step 1: Understand the Problem

The task is to add two numbers: 19.2 and -41.3. This involves combining both positive and negative values.

Step 2: Identify the Signs

19.2 is a positive number, while -41.3 is a negative number. When adding numbers with different signs, we subtract the smaller absolute value from the larger absolute value.

Step 3: Find the Absolute Values

The absolute value of 19.2 is 19.2, and the absolute value of -41.3 is 41.3.

Step 4: Subtract the Absolute Values

Subtract the smaller absolute value (19.2) from the larger absolute value (41.3): \[ 41.3 - 19.2 = 22.1 \]

Step 5: Determine the Sign of the Result

Since the larger absolute value corresponds to the negative number -41.3, the result of the addition takes the sign of this larger absolute value. So, 19.2 + (-41.3) = -22.1.

Key Concepts

Understanding Absolute Value

The absolute value of a number is all about its distance from zero on a number line, without considering its direction. Think of it as the "size" of the number but ignoring whether it's negative or positive. For example:

• The absolute value of 5 is 5.
• The absolute value of -5 is also 5.

When dealing with the absolute values of numbers, you're often just concerned with stripping away any negative signs to understand the basic magnitude of a number. In the exercise, the absolute value of 19.2 remains 19.2, and the absolute value of -41.3 is 41.3. These absolute values help us figure out how to properly combine numbers with different signs.

Working with Negative Numbers

Negative numbers are numbers less than zero and are often seen in various real-life contexts, such as temperatures below freezing. When handling negative numbers, especially in addition or subtraction, consider these pointers:

• If you're adding a negative number to a positive one, you're essentially subtracting from the positive number.
• Similarly, if you subtract a negative number, you actually add its positive counterpart.

For instance, in the exercise, when adding 19.2 and -41.3, you're subtracting the smaller number from the larger, due to the opposite signs. Always remember that the direction and sign matter in these calculations.

Grasping Basic Arithmetic with Integers

Basic arithmetic allows you to perform fundamental operations such as addition and subtraction. When dealing with integers, especially those with different signs:

• Identify which number has the greater absolute value. This dictates the final sign of your result after addition.
• Subtract the smaller absolute value from the larger absolute value when the signs differ.
• The result takes the sign of the number with the larger absolute value.

In our example, since 41.3 (absolute value of -41.3) is larger than 19.2, subtract 19.2 from 41.3, resulting in 22.1. The final answer is -22.1 because -41.3 is the number with the larger absolute value, indicating the answer will be negative.