An army of 12,000 soldiers consists of Europeans and Indians. The average height of a European soldier is 5 ft 10 inches and the average height of an Indian soldier is 5 ft 9 inches. The average height of the whole army is 5 ft 9 3/4 inches. How many Indian soldiers are there in the army?

Introduction / Context:
This problem is a classic example of weighted averages, where two groups (Europeans and Indians) with different average heights together form a larger group with a different combined average. The question asks you to work backwards from the combined average to determine the number of soldiers in one subgroup, in this case the number of Indians.

Given Data / Assumptions:

Total soldiers in the army = 12,000.
Average height of a European soldier = 5 ft 10 inches = 70 inches.
Average height of an Indian soldier = 5 ft 9 inches = 69 inches.
Average height of the whole army = 5 ft 9 3/4 inches = 5 ft 9.75 inches = 69.75 inches.
Let the number of European soldiers be E and Indian soldiers be I, with E + I = 12000.

Concept / Approach:
The combined average height of the entire army is a weighted average of the European and Indian averages, weighted by their respective group sizes. Mathematically, total height of all soldiers divided by the total number of soldiers gives the combined average. We express total height as 70 * E + 69 * I and set it equal to 69.75 * 12000. Solving this system of equations gives the required number of Indians.

Step-by-Step Solution:
Step 1: Set up the equations. E + I = 12000. Total height equation: 70E + 69I = 69.75 * 12000. Step 2: Compute the right-hand side. 69.75 * 12000 = 837000 inches. So 70E + 69I = 837000. Step 3: Eliminate I using E + I = 12000. I = 12000 - E. Substitute into height equation: 70E + 69(12000 - E) = 837000. This simplifies to 70E + 828000 - 69E = 837000. So E + 828000 = 837000, hence E = 9000. Step 4: Find I. I = 12000 - 9000 = 3000.

Verification / Alternative Check:
With 9000 Europeans at 70 inches average, total European height = 9000 * 70 = 630000 inches. With 3000 Indians at 69 inches average, total Indian height = 3000 * 69 = 207000 inches. Combined total = 837000 inches. Dividing by 12000 gives 837000 / 12000 = 69.75 inches, which is exactly 5 ft 9 3/4 inches, confirming that the numbers are consistent.

Why Other Options Are Wrong:
4000 or 5500 Indians would shift the combined average height noticeably closer to the Indian average of 69 inches, and away from the given 69.75 inches.
2700 Indians would make Europeans even more dominant, pulling the average closer than 69.75 inches to 70 inches, again contradicting the given combined average.

Common Pitfalls:
Many learners convert feet and inches incorrectly or forget to convert all heights into a single unit (like inches) before forming equations. Others try to directly average the three heights without taking the group sizes into account. Always remember that combined averages are weighted by group sizes, not simply by the number of groups.

Final Answer:
The number of Indian soldiers in the army is 3000.