Problem 68 - 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 -1,200.

Step 1: Understand the Problem

We are asked to find the difference between -222 and 987 and determine which option from the list it is closest to. This is a mental math problem, so we should simplify the operation as much as possible to identify the closest answer.

Step 2: Recall number line concepts

If you imagine a number line, subtracting a positive number from a negative number (-222 - 987) results in moving further left (deeper into negative values), because you are adding the absolute value of the positive number (987) onto the negative number (-222).

Step 3: Estimate the Difference

Without using paper or a calculator, estimate the difference: Starting at -222, think about subtracting 1,000; the resulting value is -1222. However, since we subtract 13 fewer than 1,000, we are 13 units closer to zero. Calculate -1222 + 13 to find the result.

Step 4: Compute the exact difference mentally

The actual calculation is: -222 - 987 = -1222 + 13 = -1209. Thus, the approximate value for mental math purposes can be approximated to about -1200.

Step 5: Compare with options

Now compare your result, -1209, to the options given: 200,000, 800, -800, -1,200. Observe which value is closest to -1209.

Key Concepts

Understanding Number Line Concepts

When working with number line concepts, visualize a horizontal line where numbers extend infinitely in both directions. To the right, numbers increase, while to the left, they decrease. Zero is at the center, dividing positive numbers from negative ones. When you subtract, especially with negative and positive numbers, movements are made along this number line. For example, subtracting a positive number from a negative number, as in \(-222 - 987\), involves moving further to the left, deeper into the negatives.

• Start Point: Begin at -222.
• Movement: Subtracting 987 means moving 987 steps left from -222.

This visual can help you understand why subtracting a larger positive from a negative results in a more negative value. Therefore, estimating becomes about visualizing shifts on this line.

Estimating Differences

Estimating differences enables easier mental calculations, particularly without writing or using a calculator. Start by rounding each number to a nearby, manageable value that is easier to manipulate. In our examples:

• Initial Values: -222 and 987
• Rounded: -222 is close to -200, and 987 can be approximated as 1,000.

Use these estimates to simplify your subtraction. When subtracting 1,000 from -200, the result is approximately -1,200. Though this isn’t precise, it offers a ballpark figure helpful for mental calculations. Always refine estimates by accounting for discrepancies, such as actual differences (i.e., adjusting for the 13 subtracted extra in the given problem).

Working with Negative Numbers

Negative numbers can initially be tricky, as they represent values less than zero. They behave in unique ways during computations:

• Addition moves left for negatives, right for positives.
• Subtracting positives from negatives increases their negative value.

Consider the likeness of negative numbers to debt: more borrowing (subtracting positives) deepens your debt (increases negativity). In the problem of \(-222 - 987\), think of taking a big step left on a number line. Understanding negative interactions helps estimate correctly and approach problems with confidence.

Understanding Integer Subtraction

Integer subtraction extends beyond regular arithmetic, particularly when involving negatives and positives. Think of subtraction as taking away or moving left.

• Subtracting Positives: Move left (become more negative).
• Subtracting Negatives: Move right or do the opposite motion (often effectively adding the positive form of that number).

Using intuitive understanding (like debts or deficits for negatives) can help solidify why operations work as they do. Practicing with negative integers and visualization allows tackling these challenges head-on. For instance, \(-222 - 987\) simplifies understanding these principles, shaping more accurate and faster mental math strategies.