Problem 43 - Prealgebra

Question

Simplify as much as possible by first changing all subtractions to addition of the opposite and then adding left to right. $$101-(-95)+6$$

Step-by-Step Solution

Answer

The simplified result is 202.

Step 1: Change Subtraction to Addition

The original expression is given as \( 101 - (-95) + 6 \). To simplify, change the subtraction to addition by recognizing that subtracting a negative number is the same as adding the positive of that number. Therefore, \( 101 - (-95) \) becomes \( 101 + 95 \). The new expression is: \( 101 + 95 + 6 \).

Step 2: Addition from Left to Right

Next, add the numbers together from left to right. First, add \( 101 + 95 \): \( 101 + 95 = 196 \).

Step 3: Final Calculation

Continue by adding the remaining number: \( 196 + 6 \). Perform this last addition: \( 196 + 6 = 202 \).

Key Concepts

Addition of Integers

When adding integers, you simply put together the values. It's like adding physical objects; for example, if you have 5 apples and gain 3 more, you now have 8 apples. The same rule applies to numbers.
\( 5 + 3 = 8 \). This is straightforward when dealing with positive numbers, but you can also add negative numbers. Remember, negative numbers represent a sort of subtraction in value.
When adding a positive integer with a negative integer, the operation results in a subtraction of the values. Imagine adding 5 and -3. This is equivalent to starting with 5 then taking away 3, which leaves you with 2. Graphically, if you do this on a number line, you move right starting from zero for the positive numbers, and left for the negative numbers.
This concept can be universally applied for whether positive or negative, simplifying complex calculations.

Subtraction as Addition of Opposites

Subtraction can often be a confusing operation, but thinking of it in terms of addition can ease the process. Subtraction boils down to adding the opposite of a number. This is central in integer operations.
For example, subtracting \(-95\) is the same as adding its opposite \(95\). Hence, the operation \(101 - (-95)\) becomes \(101 + 95\).

• This conversion makes it easier to perform the operation, especially mental arithmetic.
• Changing subtraction to addition of opposites helps maintain consistency and clarity in mathematical evaluations.
• Recognizing this pattern is useful in solving complex mathematical problems by reducing error chances.

This technique can be used universally to simplify the arithmetic involving negative numbers.

Order of Operations

Order of operations is crucial to accurately performing calculations. Mathematics follows a standardized set of rules for which operations are performed first, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
In the problem \(101 - (-95) + 6\), once we translate subtraction into addition, we proceed from left to right with the calculation.

• The first operation after conversion is \(101 + 95\), which simplifies to \(196\).
• Then, add \(6\) to \(196\) giving \(202\).

This method guarantees clarity and correctness regardless of complexity, ensuring that the arithmetic operations follow mathematical conventions consistently and reliably.