Problem 67 - Prealgebra

Question

Work Problems \(61-68\) mentally, without pencil and paper or a calculator. The difference \(-222-(-987)\) is closest to which of the following numbers? a. \(200,000\) b. 800 c. \(-800\) d. \(-1,200\)

Step-by-Step Solution

Answer

The difference is closest to option b, 800.

Step 1: Understand the Problem

To solve this problem, we need to find the difference between \(-222\) and \(-987\), which means we need to compute \(-222 - (-987)\). This is the same as \(-222 + 987\).

Step 2: Simplify the Expression

Convert the original expression \(-222 - (-987)\) into \(-222 + 987\). When you subtract a negative number, it is equivalent to adding the positive number.

Step 3: Calculate the Sum

Compute \(-222 + 987\). First, subtract the smaller absolute value from the larger absolute value: \(987 - 222 = 765\). Since 987 is greater and positive, the result of this operation is positive, yielding 765.

Step 4: Compare with Options

Assess the answer options against the result of 765. The closest option to 765 is 800.

Key Concepts

Mental Math

Solving math problems without the aid of paper, pencil, or a calculator is known as mental math. This method is highly beneficial for enhancing cognitive skills and quick problem-solving abilities. It allows you to perform calculations in your head by breaking the problem into simpler parts. For instance, if you're tasked with calculating \(-222 - (-987)\), understand first you'll be working with a smaller sequence of numbers. Focus on adjusting the problem by understanding the basic operations.

• Mental math relies heavily on concentration and practice.
• You can develop strategies for different types of problems to increase efficiency.
• This skill is particularly useful for quick everyday calculations and estimations.

Overall, mental math is a powerful tool for developing analytical thinking.

Integer Subtraction

Integer subtraction is simply finding the difference between two numbers. In the given problem, we must subtract \(-222\) from \(-987\).
When dealing with integers:

• Remember that subtracting a negative is the same as adding a positive.
• Convert subtraction problems into addition whenever possible to make calculations easier.
• Use number lines to visualize how numbers move left or right when subtracted or added.

For example, \(-222 - (-987)\) becomes \(-222 + 987\). The subtraction of the negative means the value increases, simplifying the task considerably.

Integer Addition

Integer addition combines positive and negative numbers to get a sum. Like subtraction, this can initially seem tricky due to the mix of positive and negative integers. However, a firm grasp of integer addition is essential.
The steps involve:

• Identifying whether numbers are positive or negative.
• Adding the absolute values of integers when they share the same sign.
• Finding the difference in absolute value when they have different signs, with the sign of the integer with the larger value determining the sum's sign.

In this example, adding \(-222\) to 987 is simplified by subtracting the smaller absolute value from the larger one, resulting in\(765\). Since 987 is greater, the sum is positive.

Absolute Value Comparison

Absolute value refers to the distance of a number from zero on a number line, without considering direction. It is always a positive number or zero. Absolute value helps simplify problems involving positive and negative numbers, such as integer addition or subtraction.
To use absolute values in math problems:

• Convert all numbers to their absolute values before comparing or performing operations.
• For subtraction problems, find the difference in absolute values to simplify calculations.
• In this exercise, compare and operate on 987 and 222, regardless of their original negative signs.

Thus, recognizing absolute values helps to manage complicated calculations and makes solving problems faster.