A grandmother wants to distribute all her assets worth Rs 52,500 among her three grandsons in the ratio 1/35 : 1/21 : 1/15. What is the value of the smallest share among these three portions?

Introduction / Context:
This ratio and proportion question is about distributing a fixed amount of money among three people in a given fractional ratio. The twist is that the shares are expressed as fractions, and we have to convert that information into actual rupee amounts and then identify the smallest share. Such questions are common in aptitude exams because they test comfort with fractions, ratios and basic arithmetic in a practical context involving inheritance or division of money.

Given Data / Assumptions:
- Total assets of the grandmother are Rs 52,500.
- The three grandsons share the money in the ratio 1/35 : 1/21 : 1/15.
- The ratio refers to the amounts received by each grandson respectively.
- We are asked to find the smallest of the three final rupee amounts.

Concept / Approach:
The key idea is that even if the terms of a ratio look complicated, they can be treated as proportional parts. We first convert the fractional terms 1/35, 1/21 and 1/15 into a common framework. Let the common proportional factor be k. Then the three shares are k*(1/35), k*(1/21) and k*(1/15). The sum of these three expressions must equal the total amount of Rs 52,500. Once we find k, we can compute each share and then select the smallest share among them.

Step-by-Step Solution:
Step 1: Let the three shares be k*(1/35), k*(1/21) and k*(1/15). Step 2: Add the three fractional parts: 1/35 + 1/21 + 1/15. Step 3: The sum of these fractions is 1/7, so k * (1/7) = 52,500. Step 4: Solve for k to get k = 52,500 * 7 = 3,67,500. Step 5: First grandson share = 3,67,500 * (1/35) = Rs 10,500. Step 6: Second grandson share = 3,67,500 * (1/21) = Rs 17,500. Step 7: Third grandson share = 3,67,500 * (1/15) = Rs 24,500. Step 8: Compare the three amounts: 10,500, 17,500 and 24,500. The smallest is Rs 10,500.

Verification / Alternative check:
A quick check is to confirm that the three shares add up to the original total. Add 10,500 + 17,500 + 24,500 which equals 52,500, so the distribution is consistent. A second check is to compute the ratio of the three rupee amounts and see whether it simplifies to 1/35 : 1/21 : 1/15 after multiplying by a common factor. Since the method used directly followed from the algebraic form, the result is reliable.

Why Other Options Are Wrong:
Rs 12,800 does not correspond to any of the three computed shares and would make the total distribution larger than Rs 52,500 if used as the smallest share. Rs 15,000 is larger than the true smallest share and also does not match any calculated portion. Rs 17,500 is actually the middle share, not the smallest one, so it cannot be the correct answer to the question that specifically asks for the smallest share.

Common Pitfalls:
A common mistake is to treat 1/35 : 1/21 : 1/15 as 1 : 35 : 21 : 15 or to invert the fractions accidentally. Another frequent error is to ignore the fractional nature of the ratio and attempt to convert them directly to percentages without a clear plan. Some learners also forget to ensure that the sum of the three calculated shares is equal to the total assets, which is an important consistency check.

Final Answer:
Thus, the smallest of the three shares received by the grandsons is Rs 10,500.