Problem 42 - Introductory and Intermediate Algebra for College Students 4th

Question

Find each sum without the use of a number line. $$19+(-5)+1+8+(-13)$$

Step-by-Step Solution

Answer

The sum of the numbers 19, -5, 1, 8, -13 is 10.

Step 1: Group Positive and Negative Numbers

It can be helpful to first group the positive and negative numbers together under addition. This gives: \(19 + 1 + 8 + (-5) + (-13)\).

Step 2: Add Positive Numbers

Next, add the positive numbers together: \(19 + 1 + 8 = 28\).

Step 3: Add Negative Numbers

Then, add the negative numbers together: \(-5 + -13 = -18\). Remember that adding negative numbers will give a more negative result.

Step 4: Add Results

Finally, add the results from step 2 and step 3 together. This is like a subtraction, as the second term is negative: \(28 + (-18) = 10\).

Key Concepts

Grouping Numbers

When solving problems involving a mix of positive and negative numbers, it is often helpful to organize them before performing operations. This is known as grouping numbers. By grouping all positive numbers and all negative numbers separately, you can simplify the calculation process. This method allows you to tackle each set of numbers independently, making the arithmetic clearer and easier to manage.
In our exercise, we began by rearranging the terms:

• Positive numbers: 19, 1, 8
• Negative numbers: -5, -13

This organizing step makes it simpler to perform additions within each category. It also prepares us for future steps, where we'll add each group separately before combining them. Grouping is particularly useful for students who find mixing signs confusing, as it breaks down the problem into smaller, more manageable parts.

Adding Positive Numbers

Positive numbers are straightforward when it comes to addition. There are no signs to worry about, and the process is just finding the total sum. In our example, the positive numbers are 19, 1, and 8. When adding these, you simply combine them as usual:

• First, add 19 and 1, which is 20.
• Then, add 8 to 20, resulting in 28.

This step is usually simple since it involves basic addition among positive integers. Remember, the order in which you add doesn't matter; the result will be the same, thanks to the associative property of addition.

Adding Negative Numbers

Adding negative numbers might seem a bit trickier since it involves combining numbers that all have a minus sign in front of them. However, once you understand that the result of adding negative numbers is a further subtraction, it can be more intuitive.
In our example, the negative numbers are -5 and -13. When you add these, think of moving further on the negative side of the number line:

• Add -5 and -13: this gives you -18.

Essentially, when you add a string of negative numbers, you're simply adding their absolute values and then applying a negative sign to the result. This process can alternatively be thought of as combining debts; adding more negative value increases the total debt.