Problem 76 - Prealgebra

Question

Work mentally, without pencil and paper or a calculator. The answer to the problem -251+(-249) is closest to which of the following numbers? a. 500 b. 0 c. -500

Step-by-Step Solution

Answer

The answer is closest to option c: -500.

Step 1: Understanding the Problem

We need to mentally calculate the sum of two negative numbers, -251 and -249, and determine which given option this sum closely matches.

Step 2: Adding the Negative Numbers

We add the numbers -251 and -249 together. Remember, adding two negative numbers results in a larger negative number. The equation is: \(-251 + (-249) = -251 - 249\)

Step 3: Calculating the Sum

Combining -251 and -249, we sum their absolute values: 251 + 249 = 500. The result of the addition in terms of negative values is -500.

Step 4: Comparing Results to Given Choices

The result of the sum is -500. We compare this number to the given options. - The answer is closest to option c: -500.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero. They are usually represented with a minus sign (e.g., -3, -12). Understanding how to work with negative numbers is vital for many mathematical operations. In the context of addition, when you add negative numbers, the result is a more negative number because both numbers are below zero.

• Understanding the Direction: Whenever you add a negative number, it's like moving to the left on a number line, further away from zero.
• Example: If you begin at -3 and add -4, you travel four steps left on the number line, landing at -7.

It's helpful to visualize using a number line to see how the sum of two negative numbers becomes even larger in magnitude but still negative.

Tips for Mental Math with Integers

Mental math, especially with integers, can be challenging but rewarding. It improves your number sense and ability to calculate quickly.
One great way to develop mental math skills is through estimation and using friendly numbers. Here are some strategies:

• Chunking: Break numbers into more manageable parts. For example, when adding -251 and -249, think of them as -250 and -250 to easily see the sum as -500.
• Estimation: Round numbers to the nearest ten or hundred to quickly estimate the result.
• Visualization: Use visual aids like a number line in your mind to visualize adding or subtracting.
• Pattern Recognition: Recognize patterns in numbers to simplify calculations. Notice how -251 and -249 are similar in size, indicating the sum will closely resemble twice their magnitude.

Developing these skills will improve speed and accuracy in addition, making mental math less daunting.

Using Absolute Value in Integer Addition

Absolute value is a concept that helps in understanding the magnitude of numbers without considering their sign. It is denoted by two vertical bars, like this: \(|x|\). The absolute value of a number is its distance from zero on the number line, regardless of direction.

• Understanding without Sign: The absolute value of both -3 and 3 is 3; it ignores whether the number is positive or negative.
• Application in Adding Integers: When adding negative numbers, compute the absolute values first to simplify calculations.
• Example: In our exercise, to add -251 and -249, first consider their absolute values: 251 and 249. Adding these gives 500, which then helps us quickly identify that the sum in negative terms is -500 since you simply flip the sign back.

This method not only aids in accuracy but also provides a clearer understanding of number magnitudes during calculation.