Problem 77 - College Algebra with Modeling and Visualization

Question

Calculate the percent change for the given A and B. Round your answer to the nearest tenth of a percent when appropriate. $$ A=\$ 8, B=\$ 13 $$

Step-by-Step Solution

Answer

The percent change from \$8 to \$13 is 62.5\%.

Step 1: Identify Initial and Final Values

First, determine the initial and final values. In this exercise, the initial value \(A\) is \\(8 and the final value \(B\) is \\)13. We are calculating the percent change from \(A\) to \(B\).

Step 2: Calculate the Difference

Find the difference between the final and the initial values. Subtract \(A\) from \(B\): \(\text{Difference} = B - A = 13 - 8 = 5\).

Step 3: Calculate the Percent Change

To find the percent change, use the formula: \(\text{Percent Change} = \left(\frac{\text{Difference}}{\text{Initial Value}}\right) \times 100\). Substitute in the values: \(\frac{5}{8} \times 100 = 62.5\%\).

Step 4: Round to the Nearest Tenth

Since the instruction is to round to the nearest tenth of a percent, and our calculation of \(62.5\%\) is already at one decimal place, no further rounding is needed.

Key Concepts

Initial and Final Values

Understanding what initial and final values are is critical when calculating percent change. In this context, the "initial value" is where you start, like the beginning of a journey. For instance, if you have \( A = \\(8 \), this is your initial value. It's like saying, "I began with \)8." The "final value" is your destination or where you end up. In our example, that would be \( B = \\(13 \). This is like saying, "I ended with \)13." Knowing both these values is essential because they represent the start and end points you will use to find out how much change has occurred.

Difference Calculation

Once you've identified your initial and final values, the next step is to calculate the difference between them. Start by subtracting the initial value from the final value. Hence, you use the formula:

• Difference = Final Value - Initial Value.

For our specific case, this means calculating 13 minus 8.
Thus, the difference is 5.
It's as simple as finding out how much more you have now compared to what you started with. This difference is what you'll use to determine the percent change.

Thinking of difference as the change in the amount you have between two points in time helps set the foundation for percent change calculations.

Percent Rounding

After calculating the percent change, the final step is to round appropriately if asked. Rounding means simplifying the number to a particular decimal place for easier interpretation.
First, calculate the percent change using the formula:

• \(\text{Percent Change} = \left(\frac{\text{Difference}}{\text{Initial Value}}\right) \times 100\).

In our case, 5 divided by 8, multiplied by 100, equals 62.5%.
Since the instruction is to round to the nearest tenth of a percent, and our answer is already at one decimal place, 62.5% requires no further rounding.
Rounding is mostly necessary when your percent change has more decimal places than requested. This step ensures the final result is both accurate and easy to communicate.
Use rounding only when needed to match the level of precision specified in any given problem.