Problem 73 - Beginning and Intermediate Algebra

Question

Find each difference. $$ 3.4-(-8.2) $$

Step-by-Step Solution

Answer

11.6

Step 1: Understand the Problem

The given problem is to find the difference between two numbers: 3.4 and -8.2. The expression to solve is 3.4 - (-8.2).

Step 2: Simplify the Expression

Notice that subtracting a negative number is equivalent to adding the positive version of that number. So, the expression 3.4 - (-8.2) can be rewritten as 3.4 + 8.2.

Step 3: Perform the Addition

Now, add the two numbers: 3.4 + 8.2. Using simple addition, align the decimal points and add the values: 3.4 +8.2 ------ 11.6

Step 4: Write the Final Answer

The sum of 3.4 and 8.2 is 11.6. Therefore, the difference 3.4 - (-8.2) equals 11.6.

Key Concepts

Adding Positive Numbers

When adding positive numbers, the key idea is to simply combine the values to get the total sum. Think of it like putting together two stacks of blocks. Each block from the first stack is added to each block from the second stack, and the result is a bigger stack.
For instance, if you want to add 2 and 3, you count all the blocks together. Thus, 2 + 3 = 5.
Here's a quick reminder of the rules:

• The order in which you add numbers doesn't change the result. This is known as the commutative property of addition: a + b = b + a.
• Adding zero to a number leaves it unchanged: a + 0 = a.
• If you have more than two numbers, you can group them in any way to make adding easier. This is the associative property: (a + b) + c = a + (b + c).

Just like in our exercise, where 3.4 is added to 8.2 by aligning the decimal points. Adding them results in 11.6.

Decimal Addition

Adding decimals follows the same principle as adding whole numbers, but it's crucial to align the decimal points. This means that the digits to the right of the decimal point need to line up directly above or below each other.
Let's break down the process:

• Write the numbers in a column, aligning the decimal points.
• If there are empty spaces, you can fill them with zeros to make the addition easier.
• Add the numbers column by column, starting from the rightmost digit and moving to the left.
• Place the decimal point directly below the other decimal points in your final answer.

This aligns perfectly with our example of adding 3.4 and 8.2:

3.4
+ 8.2
--------
11.6
As you can see, aligning the decimals ensures that each digit is properly added, giving us the correct sum.

Understanding Subtraction

Subtraction can sometimes seem tricky, especially when negative numbers are involved. To understand subtraction deeply, it's important to see it as the process of finding the difference between numbers.
In subtraction:

• When you subtract a smaller number from a bigger one, you find out how much more the bigger number is.
• When dealing with negative numbers, subtracting a negative is like adding a positive. This happens because removing a negative is the same as adding its positive counterpart.

Let's look at our given example, \text{\(3.4 - (-8.2)\)}.
When you see subtraction of a negative number, like in \text{-(-8.2)}, you essentially change it to adding a positive: \text{3.4 + 8.2 = 11.6}.
This is an important rule to remember:

• Subtracting a negative number is the same as adding the absolute value of that number: \text{a - (-b) = a + b}.

By turning the subtraction into an addition, we make the problem simpler to handle and arrive at the correct answer more easily.