A shepherd had n goats in the year 2000; in 2001 the number increased by 40%, in 2002 it decreased to 70% of that, in 2003 it grew again by 30%, and in 2004 he sold 10% of the goats leaving him with 34,398 goats; what was the overall percentage increase in the number of goats over this entire duration?

Introduction / Context:
This question is about successive percentage increases and decreases applied to a population of goats over several years. After a sequence of changes, a certain number of goats remain. From this final value we must infer the initial number and then compute the net percentage increase over the whole period. Such problems are excellent practice for understanding compound effects of percentages and are common in topics like population growth, interest, and depreciation.

Given Data / Assumptions:

• Initial number of goats in year 2000 is n.
• In 2001, number increases by 40% (multiplied by 1.4).
• In 2002, number decreases to 70% of that year 2001 value (multiplied by 0.7).
• In 2003, number increases by 30% (multiplied by 1.3).
• In 2004, 10% of goats are sold, leaving 90% of the previous count (multiplied by 0.9).
• After all these changes, the final number of goats is 34,398.
• We must find the overall percentage increase from n to 34,398.

Concept / Approach:
Successive percentage changes can be represented by successive multiplication factors. The overall effect is the product of these factors. If the overall factor times n gives the final number, we can find n by dividing the final number by that factor. Once we know both the initial and final values, we compute net percentage increase by the formula: (final - initial) / initial * 100. This method avoids doing year by year tracking in absolute numbers and instead uses multiplication factors efficiently.

Step-by-Step Solution:
Increase of 40% in 2001 corresponds to multiplying by 1.4.Decrease to 70% in 2002 corresponds to multiplying by 0.7.Increase of 30% in 2003 corresponds to multiplying by 1.3.Sale of 10% in 2004 corresponds to keeping 90%, so multiply by 0.9.Overall multiplier from 2000 to final is 1.4 * 0.7 * 1.3 * 0.9.Compute stepwise: 1.4 * 0.7 = 0.98, 0.98 * 1.3 = 1.274, and 1.274 * 0.9 = 1.1466.Thus final number = 1.1466 * n = 34,398.So n = 34,398 / 1.1466 = 30,000.Net increase = final - initial = 34,398 - 30,000 = 4,398 goats.Percentage increase = (4,398 / 30,000) * 100 = 14.66%.

Verification / Alternative check:
We can verify by moving forward from 30,000 goats year by year. After 40% increase: 30,000 * 1.4 = 42,000. After 2002 decrease to 70%: 42,000 * 0.7 = 29,400. After 30% increase in 2003: 29,400 * 1.3 = 38,220. After selling 10% in 2004, remaining goats are 38,220 * 0.9 = 34,398. This matches the given final number perfectly. Calculating net increase again: increase is 4,398 goats on 30,000, giving a percentage of 4,398 / 30,000 * 100 = 14.66%, consistent with the earlier calculation.

Why Other Options Are Wrong:
16.66% and 20% are higher than the correct value and would require a larger final number relative to 30,000.

33.33% implies a much larger growth factor that does not match the sequence of mixed increase and decrease percentages provided in the problem.

Common Pitfalls:
One common mistake is to simply add and subtract percentages, for example 40% minus 30% minus 10% and so on, rather than multiplying the appropriate factors. Another mistake is to treat the decrease to 70% as a 30% decrease on the original 30,000 instead of the intermediate value. To avoid these errors, write each year change as a clear multiplication factor and then multiply them. When the final value is given, always solve back for the initial and compute the net percentage correctly.

Final Answer:
The overall percentage increase in the number of goats over the entire duration is 14.66%.