Amit donates 20% of his monthly income to a school and then deposits 20% of the remaining amount in his bank account. After these two actions, he is left with Rs. 12,800. What is Amit's total monthly income in rupees?

Introduction / Context:
This is a typical percentage and income-based word problem. Amit spends part of his income on donation and savings in a bank. The problem describes two consecutive percentage deductions from his initial income and tells us how much money remains afterwards. We must work backwards to find his original monthly income. Such questions test understanding of successive percentage operations and the idea of working in reverse from a final amount.

Given Data / Assumptions:

• Let Amit's monthly income be I rupees.
• He donates 20% of I to a school.
• He then deposits 20% of the remaining amount in his bank.
• After donation and deposit, he is left with Rs. 12,800.
• We must find the value of I.

Concept / Approach:
The key idea is to carefully track the remaining percentage of income after each operation. First, donating 20% leaves 80% of the income. Then, depositing 20% of that remainder leaves 80% of the remainder. Overall, he keeps 80% of 80% of his original income, which is 64% of I. This final amount is given as Rs. 12,800, so we set 0.64I = 12,800 and solve for I. This approach is much faster and less error-prone than guessing values.

Step-by-Step Solution:
Let Amit's monthly income be I rupees. He donates 20% of I, so donation = 0.20I. Amount left after donation = I − 0.20I = 0.80I. He deposits 20% of the remaining amount in his bank. Deposit amount = 20% of 0.80I = 0.20 * 0.80I = 0.16I. Amount left after deposit = 0.80I − 0.16I = 0.64I. We are told that this remaining amount equals Rs. 12,800. So, 0.64I = 12,800. Therefore, I = 12,800 / 0.64. 0.64 = 64 / 100, so I = 12,800 * 100 / 64. Compute 12,800 / 64 = 200. Thus, I = 200 * 100 = Rs. 20,000.

Verification / Alternative check:
Starting from Rs. 20,000, donation at 20% is 0.20 * 20,000 = Rs. 4,000. Remaining amount = 20,000 − 4,000 = Rs. 16,000. Deposit is 20% of 16,000, which equals 0.20 * 16,000 = Rs. 3,200. Amount left after deposit = 16,000 − 3,200 = Rs. 12,800, which matches the amount stated in the question. Hence the calculated income is correct.

Why Other Options Are Wrong:
18,000 and 16,000: These would leave significantly less than Rs. 12,800 after deducting the two percentages.
24,000 and 32,000: These would leave more than Rs. 12,800 after operations, and do not satisfy the equation 0.64I = 12,800.

Common Pitfalls:
A common error is to subtract 40% directly from the income and assume that the remainder is 60%, ignoring that the second 20% is taken on the already reduced amount, not on the original income. Another mistake is to add the percentages incorrectly. Always express successive percentage changes as multipliers (like 0.80 * 0.80) and then apply them to the original amount or work backwards from the final amount to the original.

Final Answer:
Amit's total monthly income is Rs. 20,000.