Problem 39 - Prealgebra

Question

Simplify as much as possible by first changing all subtractions to addition of the opposite and then adding left to right. $$12-30-47$$

Step-by-Step Solution

Answer

The simplified expression is \(-65\).

Step 1: Change Subtractions to Additions

The expression given is \(12 - 30 - 47\). In this step, change each subtraction to adding the opposite. For \(12 - 30\), change to \(12 + (-30)\). Similarly, for \(-47\), change it to \(+ (-47)\). The expression becomes \(12 + (-30) + (-47)\).

Step 2: Add Left to Right

Now, we simply add from left to right. Start with the first two numbers: \(12 + (-30) = -18\) because adding a negative is equivalent to subtraction. Next, add the result to the last number: \(-18 + (-47) = -65\).

Step 3: Conclusion of Simplification

By adding left to right, we have simplified the expression to a single number. Thus, \(12 - 30 - 47\) simplifies to \(-65\).

Key Concepts

Expression Simplification

Expression simplification is all about making a math problem easier to think about or solve. Imagine having a backpack filled with lots of items, and you'd like to make it lightweight by removing unnecessary things. That's very much like what we do in math with simplification!

To simplify an expression, like the one given: \(12 - 30 - 47\), our first job is to rewrite it in a way that makes the arithmetic straightforward. Here, simplification involves changing each subtraction to an addition of the opposite number. Think of subtraction as just adding a negative number; it's easier to process, and it helps maintain consistency when performing arithmetic operations.

The primary goal is to reduce the expression to its simplest form by applying these simplification techniques. Remember, simplified expressions make it easier to analyze and solve problems, saving time and minimizing errors.

Addition of Opposites

The addition of opposites relies on a neat trick in arithmetic where subtraction can conveniently be turned into addition. Instead of just taking away, we add the opposite. Here's how:

• Every subtraction can be changed to addition by simply using the opposite sign.
• For example, instead of doing \(12 - 30\), think of it as \(12 + (-30)\).
• Likewise, \(-47\) gets rewritten as \(+ (-47)\).

Changing subtractions into additions makes the arithmetic consistent and more manageable.

This approach helps to streamline calculations because addition is often more intuitive and uniform compared to subtraction. By adding the opposite, we also eliminate confusion and reduce the number of operations we have to keep track of. It turns everything into an addition problem, which simplifies further calculations.

Integer Arithmetic

Integer arithmetic involves working with whole numbers, which can be either positive or negative. In problems like simplifying expressions, understanding integer arithmetic is vital. Here's why:

• When you add a positive integer to a negative integer, or vice versa, it’s like finding the difference between the numbers, keeping the sign of the larger in absolute value.
• For instance, in \(12 + (-30)\), since 30 is larger, the result is \(-18\).
• Continuing this concept, \(-18 + (-47)\) means we're simply adding two negatives, resulting in a more negative number: \(-65\).

Understanding these basic rules helps us efficiently navigate through and simplify expressions involving integers. It's about knowing how positive and negative numbers interact seamlessly, getting to the right solution without getting tangled in overly complex calculations.