Problem 72 - Elementary and Intermediate Algebra

Question

Add. $$ 33.12+(-35.7)+2.98 $$

Step-by-Step Solution

Answer

The sum is 0.40.

Step 1: Rearrange the Expression

First, look at the expression and note that it involves adding a positive number, a negative number, and another positive number: \[ 33.12 + (-35.7) + 2.98 \]We can rearrange it for clarity, though not necessary in this case, retain the original order for structured solution.

Step 2: Add the First Numbers

Start by adding the first two numbers: \[ 33.12 + (-35.7) \]This is equivalent to subtraction because adding a negative number is the same as subtracting the positive counterpart. Calculate:\[ 33.12 - 35.7 = -2.58 \]

Step 3: Add the Result with the Last Number

Now take the result from the previous step and add it to the remaining number in the expression:\[ -2.58 + 2.98 \]Perform the addition just like adding two numbers:\[ (-2.58) + 2.98 = 0.40 \]

Step 4: Write the Final Answer

The final result of the addition is:\[ 0.40 \]Ensure the result is precise by confirming each step and calculation was correctly followed.

Key Concepts

Addition of Real Numbers

When it comes to adding real numbers, which include both positive and negative numbers, the main principle is always moving along a number line. Real numbers can be integers, fractions, or decimals. They comprise all numbers that can take any value along the number line. Importantly, this also includes zero.
In the example provided, you are performing the addition of \[33.12 + (-35.7) + 2.98\]. The term "addition of real numbers" suggests you'll be summing these values sequentially. However, when negative values enter the mix, things get a bit more complex. But don't worry, it's straightforward if you follow the rules we'll outline in the next section.
Understanding the movement along the number line helps visualize solving such expressions, keeping operations like commutative and associative properties of addition in mind. Commutative property allows rearranging numbers when adding, whereas associative aids in grouping, though in this case, they aren't necessarily used. Ultimately, your goal is to harmonize all values to arrive at the final sum.

Negative Numbers

Negative numbers bring a unique twist to arithmetic operations. These are numbers less than zero and further to the left on the number line. When adding negative numbers, think of subtracting the magnitude of the number from the positive part.
For instance, in the exercise, you have the expression \[33.12 + (-35.7)\]. Adding \(+(-35.7)\) is the same as subtracting \(35.7\): \[33.12 - 35.7\]. This results in moving leftward along the number line.
A helpful tip is to recall that adding a negative number will always decrease your total. Just imagine owing more money out when you add a negative.”
When adding another positive number, \(2.98\), you reverse the direction to the right on the number line, adjusting our total once again. Understanding negative numbers in this light helps make their role in broad arithmetical operations less intimidating.

Arithmetic Operations

Arithmetic operations form the basis of everyday math calculations. They include addition, subtraction, multiplication, and division. Mastering these operations with various numbers, including those that are positive, negative, large, or small, is essential in performing more sophisticated math problems.
In your exercise, the focus is primarily on addition and understanding how this operation interacts with negative numbers. Initially, you dealt with \(33.12 + (-35.7)\), causing a subtraction effect. Then, the sum becomes \(-2.58\), using straightforward arithmetic rules. Following this step, you added \(2.98\), straightforwardly adjusting the total.
Remember, when handling arithmetic operations, keeping track of signs and magnitudes is crucial. Always consider the order of operations, even in seemingly simple additions like this one, to get the correct solution. This ability will assist in ensuring precision and accuracy in all your future mathematical endeavors.