Understanding Percentage Changes: Net Increase or Decrease After Addition and Subtraction

This article delves into the concept of percentage changes and how they are affected by adding and subtracting a certain percentage of a number. We will explore the steps to determine the net percentage change when a number is first increased and then decreased, and vice versa. This is particularly useful in fields like Social Studies, where understanding numerical changes is crucial for making informed decisions and analyzing data.

Manipulating a Number by Percentage: Steps and Analysis

Let's start by understanding the process of manipulating a number, N, by a given percentage. Consider a mathematical problem: if a number is increased by 23% and then decreased by 23%, what is the net percentage change in the number?

Initial Calculation

Let N be the original number.

Increasing N by 23% results in: N 0.23N 1.23N. Decreasing the new value by 23% results in: 1.23N - 0.23(1.23N) 1.23N - 0.2829N 0.9471N.

Comparing the final value (0.9471N) to the original value (N), we see a net decrease:

Net change 0.9471N - N -0.0529N.

Net decrease percentage -0.0529N / N * 100 -5.29%.

Hence, the net result is a decrease of approximately 5.29%.

General Formula and Analysis

Let N1 and N2 denote the initial and final numbers respectively. Then we can derive:

N2 1 - 24/100 * 1 - 24/100 * N1 31/25 * 19/25 * N1 589/625 * N1 1 - 36/625 * N1 1 - 36/625 * N1 * 100 1 - 5.76/100 * N1.

Hence, the net percentage decrease is 5.76 when a number is increased by 24 and then decreased by 24.

Real-World Application

Let's consider a practical example:

Example: If a number is increased by 24 and then decreased by 24, what is the net percentage change?

Solution: Let the number be 100.

Number increased by 24 is 100 * 1.24 124. Increase number by 24 is then decreased by 24, so 124 * 0.76 94.24. Change in the original number 94.24 - 100 -5.76. Percentage change in the original number -5.76 / 100 * 100 -5.76% or a net decrease of 5.76%.

Detailed Step-by-Step Calculation

Let N be 100.

10 increase of N 100 * 1.1 110.

10 of 110 0.1 * 110 11.

AFTER 10 DECREASE: 110 becomes 110 - 11 99.

99 (100) decrease 100 - 99/100 * 100 100 - 99 1.

Hence, the decrease is 1.

Conclusion

Understanding the net change, especially in percentage terms, is essential in many real-world applications, from finance to social studies. Whether you are increasing or decreasing a number by a certain percentage and then performing the reverse operation, the key lies in recognizing the underlying mathematical principles.

Remember, the net change often results in a decrease, if the percentage added is not offset completely by subtracting the same or a higher percentage. This principle is crucial for accurate data analysis and decision-making in a myriad of contexts.