Problem 120 - Beginning and Intermediate Algebra

Question

Solve each problem. Both Alabama and Ohio are projected to lose 1 seat in the U.S. House of Representatives in 2020 . The states projected to gain the most seats are Texas with 4 and Florida with \(2 .\) Write a signed number that represents the algebraic sum of these changes. (Data from Election Data Services.)

Step-by-Step Solution

Answer

The algebraic sum of the changes is 4.

Step 1: Understand the Problem

First, identify the changes in the number of seats. Alabama and Ohio are losing seats while Texas and Florida are gaining seats.

Step 2: Assign Signed Numbers

Assign negative numbers to represent the loss of seats and positive numbers for the gain of seats. So, Alabama: -1, Ohio: -1, Texas: +4, Florida: +2.

Step 3: Write the Sum Expression

Write the algebraic sum of these changes: -1 + (-1) + 4 + 2.

Step 4: Perform Addition

Add the numbers step by step: -1 + (-1) = -2, -2 + 4 = 2, 2 + 2 = 4.

Step 5: Conclusion

The algebraic sum of the changes is 4.

Key Concepts

signed numbers

Signed numbers are numbers that are either positive or negative.
Positive numbers, like +4, indicate a gain or an increase.
On the other hand, negative numbers, such as -1, represent a loss or a decrease.
It's important to always use the correct sign to reflect the real-world situation accurately.
For instance, in our exercise, Alabama and Ohio are losing seats, so we use -1 for both.
Conversely, Texas and Florida are gaining seats, hence we use +4 and +2 respectively.
Understanding signed numbers is crucial because they show direction and value, simplifying the problem-solving process.

algebraic sum

The algebraic sum is the total value when adding or subtracting signed numbers.
It tells you the net effect of all the changes combined.
In our exercise, to find the algebraic sum, we combined the changes in seats for Alabama, Ohio, Texas, and Florida.
We wrote it as: -1 + (-1) + 4 + 2.
Adding up the negative values together and then the positive ones, we get -2 + 6, which equals 4.
This algebraic sum correctly shows that despite some states losing seats, the net change in total is a gain of 4 seats.

addition of integers

Adding integers involves combining positive and negative numbers.
When you add negative integers, it’s like subtracting the values.
For instance, -1 + (-1) results in -2.
When you add positive integers, they simply increase the total.
In our exercise, after calculating the negative sum as -2, we added the positive number 4, giving us a result of 2.
Finally, we added another positive number 2 which resulted in an overall sum of 4.
This careful step-by-step method ensures accuracy in calculating the final result.
Knowing how to handle these operations helps in many real-life situations involving gains and losses.