Problem 56 - Prealgebra

Question

Work Problems \(55-60\) mentally, without pencil and paper or a calculator. The answer to the problem \(-52-49\) is closest to which of the following numbers? a. 100 b. 0 c\(-100\)

Step-by-Step Solution

Answer

The answer is closest to -100.

Step 1: Understanding the Problem

The problem asks us to find a number closest to the result of the expression \(-52 - 49\). We need to evaluate this expression mentally and compare it to the given options: 100, 0, or -100.

Step 2: Mentally Calculating the Sum

First, consider the expression \(-52 - 49\). Recall that subtracting a number is the same as adding its negative. Therefore, this can be rewritten as \(-52 + (-49)\).

Step 3: Combining the Numbers

Now, combine the numbers: \(-52 + (-49) = -(52 + 49)\). Calculate mentally: 52 + 49 equals 101. Therefore, \(-(52 + 49) = -101\).

Step 4: Comparing to Given Options

Now, compare the result, \(-101\), with the options provided: 100, 0, and \(-100\). \(-101\) is closest to \(-100\).

Key Concepts

Understanding Negative Numbers

In mathematics, negative numbers can be a little tricky at first, but once you understand them, they open up a whole new world of possibilities. A negative number is simply a number that is less than zero. It is represented with a minus sign (-) in front of it.
For example:

• -3, which is 3 units below zero on the number line.
• -52, which is an even greater distance below zero.

Negative numbers are essential in representing values below a baseline, such as temperatures below freezing or debits in a bank account. These numbers are just as real as positive numbers and can be added, subtracted, multiplied, or divided in mathematical operations.
When you encounter a negative number, just remember it represents a value going in the opposite direction from positive numbers. If you think of a number line, moving to the left from zero takes you into negative territory.

Mastering Addition of Integers

The addition of integers, especially when they involve negative numbers, follows specific rules that make calculations easier.
When you add two negative numbers, the result is also a negative number. This is because you are essentially moving further away from zero. For example, if you add -52 and -49, you are combining two distances below zero, resulting in an even deeper negative:

• Think of adding -52 + -49 as combining them to get -(52 + 49), which results in -101.

This can seem counterintuitive at first, but with practice, you'll get the hang of it. A helpful tip is to consider negative numbers as representing depth or deficit, where combining them increases the depth.
As you practice mental math, visualizing integer addition on a number line can also help solidify your understanding. You'll find that with time, adding integers will become second nature.

Skills for Comparing Numbers

Comparing numbers is a fundamental skill in math, which involves determining their relative size. This skill is particularly important when dealing with both positive and negative integers.
When comparing negative numbers, remember this simple rule: a negative number is always less than a positive number, and among negative numbers, the one with the larger absolute value is actually smaller.
For example, -101 is less than -100 because it represents a greater distance from zero. In contrast:

• -1 is greater than -2 because it's less negative and closer to zero.

Comparing numbers is akin to measuring distances on a number line, where numbers further to the right are always bigger than those to the left. This rule helps in quickly determining which number is greater or smaller, easing mental calculations during exercises.
By mastering this skill, you will enhance your problem-solving abilities, allowing you to solve math problems with greater accuracy and confidence.