Before a battle, the ratio of captains to soldiers in an army was 2 : 7. During the war, 25 captains and 100 soldiers were martyred, and after these losses the new ratio of captains to soldiers became 3 : 10. How many soldiers were there in the army after the war?

Introduction / Context:
Questions on captains and soldiers are classic ratio and proportion word problems used in aptitude exams. Here, we are told the initial ratio of captains to soldiers in an army, then given how many of each are lost in battle, and finally given the new ratio after the losses. Our task is to apply basic algebra on ratios to find the number of soldiers remaining after the war. This problem tests understanding of how to translate ratio information into equations and how to handle increases or decreases in the quantities while preserving the ratio relationships.

Given Data / Assumptions:

• The initial ratio of captains to soldiers is 2 : 7.
• During the war, 25 captains are martyred.
• During the war, 100 soldiers are martyred.
• After the war, the new ratio of captains to soldiers is 3 : 10.
• We assume all quantities are whole numbers and ratios represent actual counts in the same proportion.

Concept / Approach:
The key idea is to represent the initial numbers of captains and soldiers in terms of a common multiplier using the given ratio 2 : 7. Then we subtract the numbers lost in the war from each group. The resulting numbers must satisfy the new ratio 3 : 10. This produces a simple linear equation in the unknown multiplier. Solving this equation gives us the actual original counts and therefore the final number of soldiers after the war.

Step-by-Step Solution:

Verification / Alternative check:
Using k = 50, initial captains = 2 * 50 = 100. After war, captains = 100 - 25 = 75. Soldiers after war = 350 - 100 = 250. Check ratio: 75 : 250 = 3 : 10 (since 75/250 = 3/10). Therefore, the value is consistent and confirms that 250 soldiers remained after the war.

Why Other Options Are Wrong:

• 200: This would give the ratio of captains to soldiers as 75 : 200 = 3 : 8, not 3 : 10.
• 150: This would result in 75 : 150 = 1 : 2, which is not equal to 3 : 10.
• 100: This gives 75 : 100 = 3 : 4, also not matching 3 : 10.

Common Pitfalls:
Students may incorrectly set up the equation by mixing initial and final numbers or by directly using 2 : 7 and 3 : 10 without including the losses. Another common mistake is to forget to subtract the martyred captains and soldiers before applying the second ratio. Some learners also attempt to guess values instead of forming a proper equation, which can lead to errors for less obvious numbers.

Final Answer:
The number of soldiers remaining in the army after the war is 250.