Problem 58 - Prealgebra

Question

Work Problems \(55-60\) mentally, without pencil and paper or a calculator. The answer to the problem \(-52-(-49)\) is closest to which of the following numbers? a. 100 b. 0 c. \(-100\)

Step-by-Step Solution

Answer

The answer is closest to 0.

Step 1: Change Minus to Plus

When you subtract a negative number, it is the same as adding its positive. So, \[-52 - (-49)\] becomes \[-52 + 49\].

Step 2: Perform the Addition

Now, you simply need to perform the addition:Start with the number \(-52\) and add \(49\) to it:\(-52 + 49 = -3\).

Step 3: Estimate the Answer

Finally, determine which of the given options is closest to \(-3\). The possible answers are 100, 0, and \(-100\). Clearly, \(-3\) is closest to 0.

Key Concepts

Subtraction of Negative Numbers

When dealing with subtraction of negative numbers, a common mistake is to overlook the effect of the negative sign. Here is a simple way to understand it: subtracting a negative is like adding a positive.
This is because the two negatives—subtraction and the negative sign of the number—cancel each other out.
So, if you see an expression like

• \(-52 - (-49)\),

this can be rewritten as

• \(-52 + 49\)

Always remember that subtracting a negative number involves a transformation into addition. This change in operation simplifies the arithmetic and avoids confusion.

Addition of Integers

Adding integers is all about joining two or more numbers to find their sum. When dealing with both positive and negative integers, you should focus on the signs.
Here are quick tips:

• If you add two numbers with the same sign, you just add their absolute values (ignore the sign), then give the result the same original sign.
• If you add one positive and one negative number, subtract the smaller absolute value from the larger absolute value. The sign of the number with the larger absolute value should be used for the result. For example:\(-52 + 49\)
  Subtract 49 from 52, which gives you 3. Since 52 is larger and negative, the answer is \(-3\).

You can always count on integer addition as a straightforward action that just needs a careful look at the signs.

Estimation Techniques

Estimation is a crucial skill, especially when performing mental calculations. It involves roughly determining the answer without calculating every detail.
We often use rounding or comparing to known values:

• In this exercise, the number \(-3\) is the result of the arithmetic operation. By comparing it to the given choices—100, 0, and \(-100\)—you can estimate which it is closest to without precise calculation.
• Here, \(-3\) is clearly closest to 0, as it is almost there on the number line.

Such estimation saves time and simplifies problem-solving when exactness is not critical, making it an essential part of mental mathematics.