Dileep starts a business with a capital of Rs. 8000. After six months, Narender joins him with some capital. At the end of one year, both partners receive equal amounts of profit. How much capital does Narender invest in the business?

Introduction / Context:
This question is a classic partnership problem where one partner invests at the beginning and another partner joins later. The important concept is that profit in a partnership is distributed in proportion to capital multiplied by the time for which it is invested. The condition that both partners receive equal profit allows us to set up an equation and compute the missing investment.

Given Data / Assumptions:

• Dileep invests Rs. 8000 at the start of the business.
• After six months, Narender joins the business with an unknown capital amount.
• Total business duration considered is 12 months (one year).
• At the end of the year, both Dileep and Narender receive equal shares of the profit.
• Profits are shared in proportion to capital multiplied by time of investment.

Concept / Approach:
The key idea is to use capital time units. Dileep is in the business from the beginning, so his capital works for the entire 12 months. Narender joins after 6 months, so his capital works only for the remaining 6 months. If their profit shares are equal, the products of capital and time for both must be equal. We can then form an equation involving Narender's capital and solve for it directly using simple algebra.

Step-by-Step Solution:
Step 1: Dileep's capital is Rs. 8000 and he remains in the business for 12 months. Step 2: Therefore, Dileep's capital time units are 8000 * 12 = 96000. Step 3: Let Narender's capital be N rupees. He joins after 6 months, so he is in the business for 6 months. Step 4: Narender's capital time units are N * 6. Step 5: Since their profits are equal, capital time units must be equal. Thus, 96000 = 6N. Step 6: Solve for N by dividing both sides by 6: N = 96000 / 6 = 16000. Step 7: Therefore, Narender invests Rs. 16000.

Verification / Alternative check:
Once we have N = 16000, recheck the capital time units. Dileep has 8000 * 12 = 96000. Narender has 16000 * 6 = 96000. Since both have exactly the same capital time product, their shares in the total profit must be equal. This perfectly matches the condition given in the question, which confirms that the computation is correct.

Why Other Options Are Wrong:
If Narender invested Rs. 12000, his capital time units would be 12000 * 6 = 72000, which is less than Dileep's 96000, so he would receive less profit. If he invested Rs. 24000, his capital time units would be 144000, and he would receive more profit than Dileep. Amounts such as Rs. 32000 or Rs. 8000 likewise do not equate the capital time units and so cannot produce equal profit shares. Only Rs. 16000 ensures equality between the two partners.

Common Pitfalls:
Some learners wrongly compare only capital ratios without considering the different time periods of investment. Others mistakenly assume that equal profits imply equal capitals, ignoring the fact that time also affects profit share. It is also easy to misread the problem and think that Narender joins at the very end instead of after six months. Always remember to form capital time products before comparing partnership shares.

Final Answer:
Narender's investment must be Rs. 16000, which is given in option B.