Three brothers divide Rs. 1,620 among themselves in such a way that the share of the second brother is equal to 5/13 of the combined shares of the first and third brothers. What is the share of the second brother in rupees?

Introduction / Context:
This is a ratio and sharing problem in which the share of one person is given as a specific fraction of the combined shares of the other two. The total amount is known, and we are asked to find the exact share of the second brother. This is a standard problem that reinforces algebraic modeling of relationships between quantities.

Given Data / Assumptions:

• Total amount shared among the three brothers is Rs. 1,620.
• Let the shares of the first, second and third brothers be A, B and C respectively.
• The share of the second brother is 5/13 of the sum of the shares of the first and third brothers.
• We have to determine B, the share of the second brother.

Concept / Approach:
We can represent the relationship B = (5/13)(A + C). Then we know that the total sum A + B + C equals Rs. 1,620. By expressing A + C in terms of B, we can substitute into the total and solve for B directly. This approach is efficient and avoids unnecessary introduction of extra variables.

Step-by-Step Solution:
Step 1: Let the shares of the three brothers be A, B and C.Step 2: Given that B = (5/13)(A + C).Step 3: From this, we get A + C = (13/5)B.Step 4: Total amount shared is A + B + C = 1,620.Step 5: Substitute A + C = (13/5)B into the total: (A + C) + B = (13/5)B + B.Step 6: Simplify: (13/5)B + B = (13/5)B + (5/5)B = (18/5)B.Step 7: Thus (18/5)B = 1,620.Step 8: Multiply both sides by 5: 18B = 1,620 * 5 = 8,100.Step 9: Divide by 18: B = 8,100 / 18 = Rs. 450.

Verification / Alternative check:
We can perform a quick check by computing A + C = (13/5)B = (13/5) * 450 = 13 * 90 = 1,170. Then A + B + C = (A + C) + B = 1,170 + 450 = Rs. 1,620, which matches the given total. The relation B = (5/13)(A + C) also holds because (5/13) * 1,170 = 5 * 90 = 450, which is exactly B.

Why Other Options Are Wrong:
Rs. 1,170 is actually the combined share of the first and third brothers, not the share of the second. Rs. 540 and Rs. 500 do not satisfy the relation B = (5/13)(A + C) when recomputed with the total of Rs. 1,620. 'None of these' is not correct because Rs. 450 perfectly satisfies all given conditions.

Common Pitfalls:
Some candidates misread the relationship and assume B is 5/13 of the total amount rather than of the sum of the other two shares. Others try to assign arbitrary values to A and C without properly using the equation. Always express the relationship algebraically and use the given total to solve for the required share.

Final Answer:
The share of the second brother is Rs. 450.