## Adding & Subtracting Groups of Integers

## Introduction

You should now have a good idea of how to add or subtract a pair of integers. If you haven’t seen the lesson on adding integers or subtracting integers, it is recommended that you do that before attempting to tackle groups of integers.

An *integer* is a whole number (positive or negative) or zero. Basically, integers are the same as the *counting numbers* you have used throughout elementary school. What makes them different is that they include *negative numbers.* Now instead of assuming that adding a number must increase the value of an answer, you must consider that adding a negative number actually decreases the value of an answer.

Adding two integers can be shown using a number line. For example, consider the addition problem -5 + 8:

In the problem above, the number line represents the problem well. -5 can be represented by moving five to the left and + 8 by moving 8 to the right. The answer is 3.

When adding three or more integers, the number line can get confusing. The following addition problem includes four numbers.

-6 + 4 + 9 + -6

The number line here can still be used to represent the problem, but it is not as clear as in the previous example above. Four separate arrows crowds the picture, making it more difficult to follow. It may take a few seconds to see that the final answer is 1.

When adding a group of numbers, you are allowed to reorder the numbers. The problem can be simplified by reordering it into two groups: positive numbers and negative numbers.

-6 + 4 + 9 + -6

-6 + -6 + 4 + 9 = -12 + 13

The problem can be simplified even more by combining the two negatives into a single negative. In the same way, combine the two positive numbers into a single positive number.

Note that all four arrows are still present, but they are applied in a different order. This problem is now easier to follow. Most people can quickly see that the final answer is 1.

## Lesson

Adding a group of integers can be a daunting task. Consider the following problem.

There is an easy way and a hard way to do this problem. The hard way involves following the order of operations, which states that you should do the additions from left to right:

Although the actual adding isn’t very difficult above, there are four separate additions that need to be done to get to the answer. Each time you do a separate addition, there is a chance that you’ll make a mistake in the addition, mess up one or more negative signs, or simply forget to write one or more of the numbers when you bring them down for the following step.

The second way to do this problem involves grouping the numbers into two groups: a positive group and a negative group. After you find the total of each group, combine your positive total and your negative total to get your final answer.

The example above contained only *addition*. If you have any *subtraction* in a problem, there is one additional step to do before you begin computing. Since subtraction can be replaced by “adding the opposite,” you will want to “add the opposite” in the place of each subtraction sign.

Consider the problem -15 + 21 + (-30) – 41 – (-53)

The first step in the problem is to change each “minus” sign. Then, follow the steps shown above to find the answer.

An additional trick for observant problem solvers:

Consider the problem (-30) + 10 + 18 – (-20) – 31 + (-18) + 22.

Before beginning the problem, see if there is a way to quickly eliminate or “cancel out” two or more of the numbers that are being added.

In this example, the +18 and + (-18) cancel each other out and do not need to be added. The problem can then be rewritten.

(-30) + 10 + 18 – (-20) – 31 + (-18) + 22.

(-30) + 10 – (-20) – 31 + 22

(-30) + 10 + 20 + (-31) + 22

If you take a close look at the five numbers that remain, the first one is (-30) and the sum of the second and third is 10 + 20 = 30. Since -30 and 30 are opposites, their sum is zero.

(-30) + 30 + (-31) + 22

(-31) + 22

-9

The final answer is -9.

## Examples

Add or subtract each group of numbers.

(-43) + (-28) – (-20) + 80 + (-22) – (-65)

This problem is complicated as is, so change each “minus” sign into “adding the opposite” to simplify.

Group the numbers into two groups, positives and negatives.

The total of the positives is 163.

The total of the negatives is -93.

Adding these two totals together yields the final answer.

Final answer: 70.

34 + (-21) + (-71) – (-18) + 44 + (-11) – (-39)

Just as in the first example above, change each “minus” sign into “adding the opposite” to make the problem simpler.

Group the numbers into two groups, positives and negatives.

The total of the positives is 135.

The total of the negatives is -103.

Adding these two totals together yields the final answer.

Final answer: 32.

(-81) + (-44) + (-71) – (-81) + 56 – (-59)

Just as in the first example above, change each “minus” sign into “adding the opposite” to make the problem simpler. In this problem, there are opposite numbers (81 and -81) that can be cancelled out immediately.

Group the remaining numbers into two groups, positives and negatives.

The total of the positives is 115.

The total of the negatives is -115.

The totals are opposites, so the answer is zero.

Final answer: 0.