Problem 66 - Prealgebra

Question

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

Step-by-Step Solution

Answer

Option d, -1,200, is closest to the sum -1,209.

Step 1: Understanding the Problem

We need to find the sum of the numbers \(-222\) and \(-987\). Since both numbers are negative, we add their absolute values and keep the negative sign for the result.

Step 2: Adding Absolute Values

Calculate the sum of the absolute values of \(-222\) and \(-987\). \(222 + 987 = 1209\).

Step 3: Applying Negative Sign

Since both original numbers are negative, the sum \(-222 + (-987)\) will also be negative. Therefore, the result is \(-1209\).

Step 4: Comparing with Given Options

Compare the calculated sum \(-1209\) with the provided options: a. 200,000 b. 800 c. \-800\ d. \-1,200\. The closest number to \(-1209\) is option d, \-1,200\.

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero. They often appear in real-world scenarios such as debts, temperatures below zero, and elevations below sea level. Understanding how to work with negative numbers is crucial for various math operations.
When performing operations with negative numbers, keep these tips in mind:

• If you add two negative numbers, their absolute values add together, but the result stays negative.
• Adding a negative number is the same as subtracting its absolute value from a positive number.
• When handling a mix of positive and negative numbers, think of a number line. Moving left is subtracting, and moving right is adding.

Remember, the negative sign indicates direction on the number line. Understanding this concept allows for correct execution of arithmetic operations with negative numbers.

Absolute Value

Absolute value represents the distance of a number from zero on a number line, without considering its direction. It ignores the negative or positive sign of a number and focuses on the magnitude.
To find the absolute value, simply remove the negative sign, if there is one.
Here's how absolute values work:

• The absolute value of \(-3\) is \( |−3|=3\).
• The absolute value of \(5\) is \( |5|=5\).

Using absolute values helps simplify operations with negative numbers, making it easier to handle processes such as adding integers or determining the magnitude of differences.

Mental Math

Mental math is about computing calculations quickly in your head without external tools. It's a critical skill for everyday life, as it allows you to estimate and solve problems efficiently.
Here are some tips for effective mental math:

• Break down complex calculations into smaller, more manageable parts.
• Use rounding to simplify numbers temporarily, then adjust after the calculation.
• Visualize the problem using a number line or mental images to aid understanding.

Practicing mental math not only improves your number sense but also boosts confidence in solving various math problems—like quickly comparing sums without using paper.

Addition of Integers

Adding integers involves combining positive and negative numbers. Knowing how to handle positive and negative signs is key.
Some essential rules when adding integers:

• Adding two positive integers results in a positive integer.
• Adding two negative integers results in a more negative integer.
• When adding a negative integer to a positive one, think in terms of direction on a number line:
  1. If the negative number's magnitude is smaller, the result is positive.
  2. If larger, the result is negative.
  3. If equal, the result is zero.

These rules are crucial for quickly determining the result of adding integers, especially when estimates or mental math are required.