A number is first decreased by 10% and then the resulting value is increased by 10%. The final value is 10 less than the original number. What was the original number?

Introduction:
This problem focuses on successive percentage changes. It tests the understanding that decreasing and then increasing by the same percentage does not bring the value back to its original level. Instead, the net effect is a loss when the same percentage is applied down and then up. Here the information about the final difference is used to recover the original number.

Given Data / Assumptions:
An original number is reduced by 10%.
The reduced value is then increased by 10%.
The final result is 10 less than the original number.
We need to determine the value of the original number.

Concept / Approach:
When a value is decreased by p percent and then increased by the same p percent, the overall effect is a net decrease given by p^2 / 100 percent. For p = 10, the net decrease is 1%. Using this concept, we can model the final value as 99% of the original. The difference between the original and final is given as 10, which directly gives the original value by simple proportion.

Step-by-Step Solution:
Let the original number be N. After decreasing by 10%, the new value is N * (1 - 10 / 100) = N * 0.90. This reduced value is then increased by 10%. So final value = (N * 0.90) * (1 + 10 / 100) = N * 0.90 * 1.10. Compute 0.90 * 1.10 = 0.99. Therefore final value = 0.99 * N. We are told that final value is 10 less than original, so N - 0.99 * N = 10. This simplifies to 0.01 * N = 10. Hence N = 10 / 0.01 = 1000.
Verification / Alternative check:
Verify by plugging back. If the original number is 1000, after a 10% decrease we get 1000 - 10% of 1000 = 1000 - 100 = 900. Then increase 900 by 10%: 10% of 900 is 90, so final value is 900 + 90 = 990. The difference between the original 1000 and final 990 is 10, which matches the condition in the question, confirming that the original number is 1000.

Why Other Options Are Wrong:
If the original number were 500, 2000, 3000, or 4000, performing the two successive 10% changes would not produce a final value exactly 10 less than the original. Since the net decrease is 1% of the original, the difference must be 1% of the original, and only when the original is 1000 does 1% equal 10.

Common Pitfalls:
A common misconception is to assume that a 10% decrease followed by a 10% increase returns the original value. This is incorrect because the second change is applied to a reduced base. Another pitfall is to add or subtract percentages directly without computing the net effect. Remember that successive percentage changes multiply factors, they do not simply add percentages.

Final Answer:
The original number was 1000.