A shopkeeper's earnings increase by 25% in the first year but decrease by 4% in the next year, and this pattern of +25% then -4% continues alternately. After 5 years, his earnings become Rs. 72,000. What is his present yearly earning (the earning at the start of the 5-year period)?

Introduction / Context:
This question involves successive percentage changes over multiple years, alternating between an increase and a decrease. It is a typical compound percentage problem where the final value after several years is known, and we must find the starting value. Such questions appear in topics like growth and decay, salary progression, and investment returns, and they require careful handling of multiplication factors year by year.

Given Data / Assumptions:

• The shopkeeper's earning increases by 25% in the first year.
• His earning decreases by 4% in the second year.
• This pattern of +25% and -4% continues alternately for 5 years.
• After 5 years, his earning becomes Rs. 72,000.
• We are asked to find his earning at the start of this 5-year period.
• Percent changes are applied year by year to the current earning, not to the original amount each time.

Concept / Approach:
Each percentage change can be represented as a multiplicative factor. A 25% increase corresponds to multiplying by 1.25, while a 4% decrease corresponds to multiplying by 0.96. With 5 years and an alternating pattern, the sequence of factors is: 1.25 (year 1), 0.96 (year 2), 1.25 (year 3), 0.96 (year 4), and 1.25 (year 5). The overall multiplier is the product of these five factors. If we denote the initial earning as E, then E multiplied by this overall factor equals 72,000. Solving E * overall factor = 72,000 yields the initial earning.

Step-by-Step Solution:
Step 1: Factor for a 25% increase = 1 + 25/100 = 1.25.Step 2: Factor for a 4% decrease = 1 - 4/100 = 0.96.Step 3: Over 5 years with alternating pattern, the sequence of factors is 1.25, 0.96, 1.25, 0.96, 1.25.Step 4: Group the factors: (1.25 * 1.25 * 1.25) * (0.96 * 0.96).Step 5: Compute 1.25^3 = 1.25 * 1.25 * 1.25 = 1.953125, but for this question we use the combined simplified factor 1.25^3 * 0.96^2 = 1.8.Step 6: Let the initial earning be E. Then E * 1.8 = 72,000.Step 7: Solve for E: E = 72,000 / 1.8.Step 8: 72,000 divided by 1.8 equals 40,000.Step 9: Therefore, the shopkeeper's present yearly earning is Rs. 40,000.

Verification / Alternative check:
We can verify by applying the annual changes to Rs. 40,000 step by step. After year 1, earning = 40,000 * 1.25 = 50,000. After year 2, earning = 50,000 * 0.96 = 48,000. After year 3, earning = 48,000 * 1.25 = 60,000. After year 4, earning = 60,000 * 0.96 = 57,600. After year 5, earning = 57,600 * 1.25 = 72,000. This matches the final earning given in the question, confirming that Rs. 40,000 is indeed the correct starting earning.

Why Other Options Are Wrong:
If the initial earning were Rs. 45,000, then applying the same factor of 1.8 would give a final earning of Rs. 81,000, not 72,000. For Rs. 42,000, the final amount would be 42,000 * 1.8 = 75,600, again not matching. For Rs. 46,000, the final value would be 82,800. Rs. 38,000 would yield a final value of 68,400. None of these match the required final earning of Rs. 72,000, so only Rs. 40,000 is correct.

Common Pitfalls:
Some learners wrongly add or subtract the percentages linearly (for example, considering 25% - 4% = 21% per two years) instead of multiplying factors. Others may mistakenly apply all percentages to the original figure rather than successively to the updated value each year. Another common mistake is approximating decimal factors too loosely, leading to significant rounding errors. The correct approach is to use multiplication factors accurately and to apply them consecutively over each year.

Final Answer:
The shopkeeper's present yearly earning is Rs. 40,000.