P and Q invest in a business with capitals of Rs. 85,000 and Rs. 15,000 respectively. If the profit earned after 2 years is divided in proportion to their investments, in what ratio will the profit be shared between P and Q?

Introduction / Context:
This is a basic partnership question where two partners invest fixed capitals for the same period of time. The profit at the end of the period is shared in proportion to their capital contributions. We must find the ratio in which P and Q will share the profit, given their capital investments.

Given Data / Assumptions:

• P invests Rs. 85,000.
• Q invests Rs. 15,000.
• The business runs for 2 years, but both partners remain invested for the entire duration.
• Since their time of investment is the same, profit is shared directly in the ratio of their capitals.
• We must find the ratio of P's profit share to Q's profit share.

Concept / Approach:
When the time period is equal for all partners, profit sharing depends solely on capitals. Therefore, the profit ratio is simply the ratio of the capital amounts. We write 85,000 : 15,000 and simplify by dividing both numbers by their greatest common divisor to get the simplest integer ratio.

Step-by-Step Solution:
Step 1: Write the capital ratio: P : Q = 85,000 : 15,000. Step 2: Find a common factor. Both numbers are divisible by 5,000. Step 3: Divide each by 5,000: 85,000 / 5,000 = 17, 15,000 / 5,000 = 3. Step 4: Therefore the simplified ratio of profits P : Q = 17 : 3. Step 5: This ratio will be used to divide any profit amount earned by the business after 2 years.

Verification / Alternative check:
To verify, imagine a profit of Rs. 20,000. In the ratio 17 : 3, the total parts are 20. Each part is 20,000 / 20 = 1,000. P's share would then be 17 * 1,000 = 17,000 and Q's share would be 3 * 1,000 = 3,000. The division is consistent with the capital amounts, as P has invested much more and hence gets a much larger portion of the profit.

Why Other Options Are Wrong:
Ratios such as 17 : 23, 17 : 33 and 3 : 4 do not match the capital ratio 85,000 : 15,000. For example, 3 : 4 would suggest Q invests more than P, which is not true. Similarly, 17 : 23 and 17 : 33 would not reduce to 85,000 : 15,000 when scaled. Therefore these options are inconsistent with the given capitals and cannot be correct.

Common Pitfalls:
Students may attempt to include the time period even though it is the same for both partners, which is unnecessary. Another mistake is to simplify the ratio incorrectly or to divide by different factors. Always divide both capital amounts by the same largest possible number to obtain the simplest ratio.

Final Answer:
The profit will be shared between P and Q in the ratio 17 : 3.