Problem 46 - Prealgebra

Question

Simplify as much as possible by first changing all subtractions to addition of the opposite and then adding left to right. $$-300+600-(-200)$$

Step-by-Step Solution

Answer

The simplified result is 500.

Step 1: Change Subtraction to Addition of the Opposite

To start simplifying, change all subtraction operations into addition of the opposite. The expression is \[-300 + 600 - (-200)\]. Change \(-\) to \(+\) before a negative number. Thus it becomes:\[-300 + 600 + 200\].

Step 2: Add from Left to Right

Now, start adding the numbers from left to right. Begin with the first two numbers: \(-300 + 600\).Calculate: \[-300 + 600 = 300\]. Next, add the result with the remaining number:\[300 + 200 = 500\].

Key Concepts

Addition and Subtraction

Addition and subtraction are fundamental operations in mathematics. These operations are used to combine numbers or remove them from a total. When dealing with subtraction, it is helpful to reimagine it as an addition problem. This can make calculations simpler and more intuitive.

Here's how to think about it:

• Subtraction as Addition - When you see a subtraction like \( a - b \), you can think of it as \( a + (-b) \). This helps when transitioning into integer operations.
• Working Left to Right - When solving equations with multiple additions and subtractions, go from left to right. This is important for maintaining the order of operations unless parentheses indicate a different sequence of operations.

Integer Operations

Integer operations are an essential part of prealgebra and mathematics in general. Integers include whole numbers and their opposites (negatives), including zero.

When adding and subtracting integers, consider these points:

• Addition - When you add two positive numbers, you get a larger positive number. When you add two negative numbers, you get a larger negative number.
• Subtraction - Use addition of the opposite. If you subtract a larger number from a smaller one, you'll result in a negative number.This means turning \(-b\) into \(+ (-b)\) is crucial to getting the right solution when subtracting integers.
• Negative sign - Changing the sign in front of a number changes its value to its opposite. In the provided example: \(-(-200)\), the double negative becomes positive, so it's \(+200\).

Order of Operations

The order of operations is a rule that tells you the sequence in which to perform calculations. Getting this right ensures that everyone solves a problem the same way and gets the same answer.

Here's an easy way to remember the sequence:

• Parentheses - Always do operations inside parentheses first.
• Exponents - Next, solve any exponents in the expression.
• Multiplication and Division - Work these from left to right.
• Addition and Subtraction - Again, work from left to right.

In our case, no parentheses, exponents, or multiplication/division are involved. So we move directly to addition and subtraction from left to right. This highlights the importance of understanding integer operations well to simplify accurately.