A person has Rs. 2000 and wants to distribute this amount among his five children so that each child receives Rs. 30 more than the immediately younger one. Under this arithmetic progression distribution, what will be the share of the youngest child?

Introduction / Context:
This problem involves arithmetic progression and distribution of money. The father distributes Rs. 2000 among five children, with each child getting a fixed amount more than the younger one. The question tests your understanding of arithmetic sequences and how to sum them to match a total amount.

Given Data / Assumptions:
- Total money to be distributed = Rs. 2000.
- There are five children, ordered by age: youngest, then four older children.
- Each child gets Rs. 30 more than the immediately younger child.
- We need the share of the youngest child.

Concept / Approach:
The distribution forms an arithmetic progression (A.P.) of 5 terms with common difference d = Rs. 30. Let the youngest child's share be x. Then the shares will be x, x + 30, x + 60, x + 90, x + 120. The sum of these five amounts must equal Rs. 2000. We set up that equation and solve for x.

Step-by-Step Solution:
Step 1: Let the youngest child's share be x rupees.Step 2: Then 2nd child gets x + 30, 3rd child x + 60, 4th child x + 90, and the eldest x + 120.Step 3: Total sum of all 5 shares: x + (x + 30) + (x + 60) + (x + 90) + (x + 120).Step 4: Combine like terms: 5x + (30 + 60 + 90 + 120) = 5x + 300.Step 5: Given that this total is Rs. 2000, write the equation: 5x + 300 = 2000.Step 6: Subtract 300 from both sides: 5x = 1700.Step 7: Solve for x: x = 1700 / 5 = 340.

Verification / Alternative check:
Check by listing the actual amounts: youngest = 340, then 370, 400, 430, 460. Sum = 340 + 370 + 400 + 430 + 460. First add 340 + 460 = 800, 370 + 430 = 800, plus 400 gives 800 + 800 + 400 = 2000. The distribution matches the total Rs. 2000, confirming the calculation.

Why Other Options Are Wrong:
Values like Rs. 175, Rs. 260, Rs. 290 or Rs. 325 would not produce a total of Rs. 2000 when you form a 5-term A.P. with common difference Rs. 30. They typically come from arithmetic mistakes, such as mis-summing the common difference or incorrectly dividing the remaining amount by 5.

Common Pitfalls:
Some students forget that there are five children and mis-apply the difference only four times. Others subtract 30 repeatedly from 2000 instead of setting up a proper A.P. sum. Writing the full list of terms and using basic algebra to sum them is the safest and clearest method.

Final Answer:
The youngest child's share is Rs. 340.