P, Q and R enter into a partnership in the ratio 2 : 3 : 5. After 2 months, P increases his investment by 20% and Q increases his investment by 10%. If the total profit at the end of one year is Rs. 1,90,500, what amount does R receive as his share of the profit?

Introduction / Context:
This partnership question involves three partners with an initial investment ratio, followed by changes in the investments of two partners after a certain period. We must account for both the capital amounts and the time periods to determine the effective share of each partner and then compute R's share from the total profit.

Given Data / Assumptions:

• Initial investment ratio of P : Q : R is 2 : 3 : 5.
• After 2 months, P increases his investment by 20 percent.
• After 2 months, Q increases his investment by 10 percent.
• R keeps his investment unchanged for the whole year.
• Total duration is 12 months.
• Total profit at the end of the year is Rs. 1,90,500.
• We must find R's share of the profit.

Concept / Approach:
When investments change over time, we calculate effective capital time products for each segment. Investment shares in the final profit are proportional to the sum of these products. We first convert the percentage increases into new ratios for P and Q after 2 months, then combine the contributions over the full year to obtain an overall ratio including R. Finally we apply this ratio to the total profit to find R's share.

Step-by-Step Solution:
Step 1: Let the initial investments be 2x, 3x and 5x for P, Q and R. Step 2: For the first 2 months, investments are 2x, 3x and 5x. Step 3: After 2 months, P increases his investment by 20 percent, so new P investment is 2x * 1.2 = 2.4x. Step 4: After 2 months, Q increases his investment by 10 percent, so new Q investment is 3x * 1.1 = 3.3x. Step 5: R continues with 5x for the entire 12 months. Step 6: Capital time products for first 2 months: P = 2x * 2 = 4x, Q = 3x * 2 = 6x, R = 5x * 2 = 10x. Step 7: Capital time products for remaining 10 months: P = 2.4x * 10 = 24x, Q = 3.3x * 10 = 33x, R = 5x * 10 = 50x. Step 8: Total effective products: P = 4x + 24x = 28x, Q = 6x + 33x = 39x, R = 10x + 50x = 60x. Step 9: Overall ratio P : Q : R = 28 : 39 : 60. Step 10: Sum of ratio parts = 28 + 39 + 60 = 127. Step 11: R's fraction of the profit = 60 / 127. Step 12: Total profit is Rs. 1,90,500, so R's share = (60 / 127) * 1,90,500 = Rs. 90,000.

Verification / Alternative check:
Check that 1,90,500 multiplied by 60 and divided by 127 gives exactly 90,000. Now find the value of one ratio part: 1,90,500 / 127 = 1500. Then P receives 28 * 1500 = 42,000, Q receives 39 * 1500 = 58,500 and R receives 60 * 1500 = 90,000. Adding these shares gives 42,000 + 58,500 + 90,000 = 1,90,500, which matches the total profit and confirms that the ratio and R's share are correct.

Why Other Options Are Wrong:
Rs. 90,500, Rs. 87,500 and Rs. 88,900 do not correspond to 60 parts out of 127 when scaled to 1,90,500. They either break the total sum or produce non integer values for the other partners. Therefore, they are inconsistent with the calculated ratio and must be rejected.

Common Pitfalls:
A frequent mistake is to apply the percentage increases to the total profit instead of the capital. Another error is to assume that the original 2 : 3 : 5 ratio remains unchanged after the investment changes, which is not true. Always break the year into segments, compute capital time products for each segment and then add them for every partner before forming the final ratio.

Final Answer:
R receives Rs. 90,000 as his share of the profit.