In one year, the amount collected for social welfare in a village increases by 10%, and in the next year it decreases by 10%. If after these two consecutive changes the final amount is Rs. 7,920, what was the amount at the beginning, before any increase or decrease?

Introduction / Context:
This is a classic percentage change problem involving a successive increase and decrease on the same base quantity. The question describes the amount collected for social welfare in a village over two years, with a 10% increase followed by a 10% decrease. Learners must carefully understand that a 10% increase and a 10% decrease do not cancel out because the base values are different in each step.

Given Data / Assumptions:

• In year 1, the amount increases by 10%.
• In year 2, the new amount decreases by 10%.
• The final amount after two years is Rs. 7,920.
• Let the initial amount be A rupees.

Concept / Approach:
Successive percentage changes are handled by multiplying the initial amount by the corresponding percentage factors. An increase of 10% means multiplying by 1.10, and a decrease of 10% means multiplying by 0.90. Therefore, the overall factor after both changes is 1.10 * 0.90. The final amount is the initial amount multiplied by this combined factor. To recover the initial amount, we divide the final amount by this factor. This method shows that equal percentage increase and decrease produce a net decrease, not zero change.

Step-by-Step Solution:
Step 1: Let the initial amount be A rupees. Step 2: After a 10% increase, amount becomes A * 1.10. Step 3: After a 10% decrease on this new value, amount becomes A * 1.10 * 0.90. Step 4: Compute combined factor: 1.10 * 0.90 = 0.99. Step 5: Final amount is given as 7920, so A * 0.99 = 7920. Step 6: Solve for A: A = 7920 / 0.99. Step 7: 7920 / 0.99 = 8000. Step 8: Therefore, the starting amount for social welfare was Rs. 8,000.

Verification / Alternative check:
Forward check: Start with 8000. After a 10% increase, amount = 8000 * 1.10 = 8800. After a 10% decrease in the next year, amount = 8800 * 0.90 = 7920. This matches the given final amount, so our initial estimate is correct. We also see that the net effect of +10% and -10% is a 1% decrease, since 0.99 is slightly less than 1.

Why Other Options Are Wrong:
Rs. 8,800 and Rs. 8,980 are too high; if we start with them and apply the same percentage changes, the final value will exceed 7,920. Rs. 8,650 and Rs. 7,500 do not reproduce the final amount of 7,920 when the same 10% increase and 10% decrease are applied. Only Rs. 8,000 yields exactly 7,920 after the two successive percentage changes.

Common Pitfalls:
Many learners incorrectly assume that a 10% increase followed by a 10% decrease returns the amount to its original value. This is incorrect because the decrease is calculated on a higher amount. Another pitfall is adding or subtracting the percentages directly instead of multiplying the factors. It is important to remember that percentage changes act multiplicatively, not additively, over time.

Final Answer:
The amount collected at the beginning was Rs. 8,000.