A can do a work in 12 days while B can do the same work in 15 days when each works alone. They agree to undertake the work together for a total payment of ₹450. Based on their individual contributions, what will be A's share of this amount in rupees?

Introduction / Context:
This question is a combination of time and work with profit or wage sharing. A and B perform a job together and receive a total payment. Since they work at different speeds, the payment must be divided in proportion to the amount of work each actually does. You are asked to calculate A's share out of the total amount of ₹450.

Given Data / Assumptions:

A alone can finish the work in 12 days.
B alone can finish the same work in 15 days.
They work together for the entire duration until the job is completed.
They receive a total payment of ₹450 to be divided in proportion to their work contributions.
Work rates remain constant throughout the job.

Concept / Approach:
First, we compute the work rates of A and B as fractions of the job per day. Then we find the total work rate when they work together and determine each person's fraction of the total work completed. Since payment is proportional to work done, we multiply the total payment by A's fraction of the work to find A's share.

Step-by-Step Solution:
Let the total work be 1 unit. A's daily rate = 1/12 of the job per day. B's daily rate = 1/15 of the job per day. Combined daily rate = 1/12 + 1/15. LCM of 12 and 15 is 60. Convert fractions: 1/12 = 5/60 and 1/15 = 4/60. Combined rate = 5/60 + 4/60 = 9/60 = 3/20 of the job per day. Time taken together to finish the job = 1 / (3/20) = 20/3 days. Work done by A in this time = A's rate * time = (1/12) * (20/3) = 20/36 = 5/9 of the job. Work done by B in this time = B's rate * time = (1/15) * (20/3) = 20/45 = 4/9 of the job. Thus, the work ratio of A : B = 5/9 : 4/9 = 5 : 4. Total payment = ₹450, so A's share = (5 / (5 + 4)) * 450 = (5/9) * 450 = ₹250.

Verification / Alternative check:
Since the ratio of their work contributions is 5 : 4, the ratio of their payments must also be 5 : 4. Adding up parts: 5 + 4 = 9 parts. One part is 450 / 9 = 50 rupees. Therefore, A gets 5 parts, which is 5 * 50 = ₹250, and B gets 4 parts, 4 * 50 = ₹200. Both shares add up to ₹450, confirming that the division is correct.

Why Other Options Are Wrong:
Option ₹200 corresponds to B's share, not A's, and is therefore incorrect when the question specifically asks for A's share.
Option ₹240 does not correspond to a simple 5 : 4 division of ₹450 and would break the proportionality with the actual work done.
Option ₹300 would give B only ₹150, which does not fit the 5 : 4 ratio derived from the work contributions.
Option ₹225 would imply a split that is neither 5 : 4 nor any correct simplification of the work ratio.

Common Pitfalls:
Students sometimes divide the payment equally without considering differences in work efficiency. Another error is to use time ratios directly instead of converting times to work rates and finding the actual fraction of work each person does. Remember that payment shares must follow the ratio of work done, which is derived from rate multiplied by total time worked.

Final Answer:
Hence, A's share of the payment is ₹250, so the correct option is ₹250.