Problem 58 - Prealgebra

Question

Without pencil and paper or a calculator. Which number is closest to \(-151-(-49) ?\) a. \(-200\) b. \(-100\) c. 3 d. \(7,500\)

Step-by-Step Solution

Answer

The number closest to -151-(-49) is -100 .

Step 1: Identify the Operation

The problem involves the expression \(-151 - (-49)\). Here, we need to recognize that subtracting a negative number is equivalent to adding its positive counterpart.

Step 2: Convert to Addition

Change the expression \(-151 - (-49)\) to addition: \(-151 + 49\). This will simplify the arithmetic operation to directly adding these two numbers.

Step 3: Perform the Addition

Add the numbers: \(-151 + 49\). Start by negating the larger magnitude number: \(-151\). Subtract 49 from 151 (ignoring signs for now): \[151 - 49 = 102\]. Since 151 has the larger magnitude and is negative, the result is \(-102\).

Step 4: Select the Closest Option

Compare the result \(-102\) to the given options: a. \(-200\) b. \(-100\) c. 3 d. 7,500 The number closest to \(-102\) is option b: \(-100\).

Key Concepts

Subtracting Negatives

Understanding the concept of subtracting a negative number can be simplified by thinking of it as adding a positive. It's a crucial idea because it literally changes how we perform the calculation. Instead of looking at a subtraction problem with negatives in it, flip it into addition and solve it easily. This is because the negative sign serves as a double negative, turning into a positive.

For instance, in our problem \(-151 - (-49)\), instead of subtracting \(-49\), we add \(49\) to \(-151\). This transformation is essential in simplifying many math problems.

When faced with such problems:

• Always remember: Subtracting a negative is the same as adding its positive.
• Look for these opportunities to simplify your math problems quickly and efficiently.

Mastering this concept will help in tackling various problems with confidence and clarity.

Addition of Integers

Adding integers, especially when they have different signs, can be a bit tricky at first. However, once you understand the underlying principles, it becomes quite intuitive.

Let's break it down using our problem as an example: \(-151 + 49\). The numbers have different signs, one is negative and the other is positive. Here’s how you approach it:

• Identify the larger magnitude number. In this scenario, it's 151.
• Subtract the smaller magnitude (49) from the larger magnitude (151): \(151 - 49 = 102\).
• Since the larger magnitude was negative (\(-151\)), the result is also negative: \(-102\).

This method will always work when adding numbers of opposite signs – identify magnitudes, perform simple subtraction, and attach the sign of the larger magnitude number to your result.

Comparing Magnitudes

Comparing magnitudes is a way to look at the 'size' of numbers without getting tangled in their signs. This can be particularly useful when deciding which result is closer to a given number.

For our example, after adding the integers we get \(-102\). To find which option is closest to this number, you would:

• Consider the absolute values of the given options: Compare their distances from \(-102\).
• For options given as \(-200\), \(-100\), \(3\), and \(7,500\), calculate their absolute differences from \(-102\).
• The absolute distance can be found by \(|-102 - (option)\|\).

After calculating, \(-100\) has the smallest difference of \(2\), confirming it's the closest to \(-102\). By comparing magnitudes, you effectively measure proximity without letting the direction (positive or negative) complicate the process.