Problem 80 - Prealgebra

Question

Work mentally, without pencil and paper or a calculator. The sum of -77 and -22 is closest to which of the following numbers? a. -100 b. -60 c. 60 d. 100

Step-by-Step Solution

Answer

The sum of -77 and -22 is closest to -100.

Step 1: Understanding the Operation

The problem asks for the sum of -77 and -22. This means we need to add these two negative numbers together.

Step 2: Adding the Negative Numbers

When adding negative numbers, the result will also be negative. We add the absolute values: 77 and 22, giving us 99, and then attach the negative sign back, resulting in -99.

Step 3: Estimating the Nearest Option

Now, we compare -99 with the given options to find the closest value. Since -99 is very close to -100, the sum of -77 and -22 is closest to -100.

Key Concepts

Adding Negative Numbers

When you add negative numbers, it's similar to the idea of owing money. If you owe $77 and borrow another $22, you owe $99 in total. This concept helps in mental math when working with negative numbers. Here's a quick guide:

• Identify the numbers: Both are negative in this scenario.
• Add their absolute values: Ignore the negative signs temporarily. Here, we have 77 and 22, which sum up to 99.
• Reapply the negative sign: Since both numbers were negative, the result will be negative, giving us -99.

Understanding this process makes adding negative numbers a breeze in mental arithmetic. Just remember, when you combine two negatives, the sum is always more negative.

Estimating Sums

Estimating sums is about getting a quick, approximate answer. This is incredibly useful in mental math, especially when exact numbers aren't necessary.
To estimate:

• Round numbers to their nearest tens, hundreds, etc.
• Add these rounded numbers.

In our problem, -77 and -22 round to -80 and -20, respectively. Their sum, -100, is close to the exact answer we found, -99. This kind of estimation lets you quickly check your work or make quick decisions when speed is essential.

Integer Operations

Integer operations involve basic math operations like addition, subtraction, multiplication, and division with whole numbers. Here are key points for each operation with integers:

• Addition: Adding integers with the same sign results in a sum with that sign. For mixed signs, find the difference and take the sign of the larger absolute value.
• Subtraction: Think of subtraction as adding the opposite. For example, subtracting a negative number is the same as adding its positive form.

Let's apply this: The addition of -77 and -22 required us to sum up absolute values and then apply the negative sign. Once you're familiar with these operations, handling integers—whether positive or negative—becomes straightforward in any math scenario.