Problem 57 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The closest number to the sum -151 + (-49) is -200 .

Step 1: Understanding Negative Addition

When you add two negative numbers, the sum is also a negative number, and it increases in magnitude based on the absolute values of the numbers.

Step 2: Evaluate Absolute Values

Calculate the absolute values of the numbers: - The absolute value of -151 t is 151 . - The absolute value of -49 t is 49 .

Step 3: Add Absolute Values

Add the absolute values of the numbers: 151+49=200 .

Step 4: Determine Sign and Final Sum

Since both numbers being added are negative, the result will also be negative. Therefore, -151 + (-49) = -200 .

Step 5: Closest Option Evaluation

Each option represents a potential answer. The calculated sum -200 t is equal to option a: -200 .

Key Concepts

Understanding Negative Numbers

Negative numbers can be thought of as the numbers that are less than zero. They are often used to represent a decrease, such as a loss in temperature or a withdrawal from a bank account.
When we add negative numbers together, we are increasing the "debt" or magnitude of negativity. This means that the sum of two negative numbers will always be negative.
For example, if we add -151 and -49, we end up with a larger negative number, because we are adding more to the total amount of negativity. The sum in the exercise becomes more negative the larger the numbers are.

Exploring Absolute Value

Absolute value is an essential concept when working with integers, particularly negative numbers. The absolute value of a number is the distance of that number from zero on the number line, without considering its direction.
Let's consider the integers in our exercise: the absolute value of -151 is 151, and the absolute value of -49 is 49. What this means is that, despite being negative, -151 is 151 units away from zero, and similarly, -49 is 49 units away.
This makes it easy to compare and calculate, because we're focusing on the size, rather than the sign, of the number. So when adding -151 + (-49), you add their absolute values, 151 + 49, which results in 200. The sum still needs to be negative since both original numbers were negative, resulting in -200.

Employing Mental Math

Mental math is the ability to make calculations in your head without external help from tools like calculators or paper. With practice, mental math can become an incredibly useful skill, especially for simple addition and subtraction problems like the one in this exercise.
Understanding properties of numbers, such as what makes a negative number or how to find absolute value, is fundamental for performing mental math efficiently.

• Start by recognizing the numbers involved and identifying any direct relationships, such as both being negative.
• Next, simplify the computation by using absolute values to add 151 and 49 mentally to reach 200.
• Finally, remember to apply the sign back on the sum since both numbers started negative, resulting in -200.

This step-by-step approach can aid in making quick and accurate mental calculations.