Karan starts a business by investing Rs. 29,000. After 5 months, Satish joins the business with a capital of Rs. 18,000. If at the end of the year the total profit is Rs. 16,970, what is Satish's share in the profit?

Introduction / Context:
This question involves two partners where one invests from the beginning and the other joins later. The aim is to find the second partner's share of the profit using the concept of capital time products and proportional distribution of the total profit.

Given Data / Assumptions:

• Karan invests Rs. 29,000 for the full 12 months.
• Satish invests Rs. 18,000, joining the business after 5 months.
• Satish's capital remains invested for the rest of the year.
• Total profit at the end of the year is Rs. 16,970.
• We must find Satish's share of this profit.

Concept / Approach:
In partnership problems, profit sharing is proportional to capital multiplied by time. When a partner joins late, his effective investment is smaller even if his capital is significant. We calculate capital time products for both partners, form a ratio, and then apply that ratio to the known total profit to find individual shares. Sometimes this leads to fractional values that are rounded to the nearest rupee in multiple choice options.

Step-by-Step Solution:
Step 1: Karan invests Rs. 29,000 for 12 months. Step 2: Karan's capital time product = 29,000 * 12 = 3,48,000. Step 3: Satish joins after 5 months, so he invests for 7 months. Step 4: Satish's capital time product = 18,000 * 7 = 1,26,000. Step 5: The ratio Karan : Satish based on capital time is 3,48,000 : 1,26,000. Step 6: Simplify this ratio by dividing both terms by 6000: 3,48,000 / 6000 = 58 and 1,26,000 / 6000 = 21. So the ratio is 58 : 21. Step 7: Total number of ratio parts = 58 + 21 = 79. Step 8: Satish gets 21 parts out of 79 from the total profit. Step 9: Satish's share = (21 / 79) * 16,970, which is approximately Rs. 4511.

Verification / Alternative check:
Calculate the value of one ratio part as 16,970 divided by 79, which is approximately 215 rupees. Satish's 21 parts then equal about 21 * 215 ≈ 4515, and rounding and small differences due to division give the option closest to the exact fractional result as Rs. 4511. Karan's share is the remaining profit, which is close to the theoretical amount given by 58 parts out of 79. Hence Satish's share of Rs. 4511 is the best fit among the given options and matches the ratio based calculation.

Why Other Options Are Wrong:
Rs. 5422, Rs. 5489 and Rs. 6145 correspond to much larger fractions of the total profit than 21 / 79 and would imply a capital time ratio very different from 58 : 21. These values break the proportional relationship between investments and profits and do not match the calculations based on the given data.

Common Pitfalls:
A common error is to assume that Satish invests for the full year instead of accounting for the late start after 5 months, which inflates his effective investment and gives a wrong share. Another mistake is to stop at a rounded ratio without checking that the final monetary shares still sum exactly to the total profit. Always include the timing factor and review the final numbers against the total profit.

Final Answer:
Satish's share in the profit is Rs. 4511 (approximately, matching the closest option).