Question

A and B entered into a partnership, investing ₹$16000$ and ₹$12000$ respectively. After $3$ months, A withdrew ₹$5000$ while B invested ₹$5000$ more. After $3$ more months, C joined with ₹$21000$. At the end of a year, the profit was ₹26,400. By how much does B's share exceed C's share?
 

• A. 3600
• B. 2100
• C. 3000
• D. 2300

Hint:

For partnerships with changing capitals, always use capital × time to form the profit ratio.
 

Answer 1

The Correct Option is A

Use time–capital products (in ₹–months).
A: \(16000\) for \(3\) months, then \(11000\) for \(9\) months \(\Rightarrow 16000\cdot 3+11000\cdot 9=147000.\) 
B: \(12000\) for \(3\) months, then \(17000\) for \(9\) months \(\Rightarrow 12000\cdot 3+17000\cdot 9=189000.\) 
C: joins at month \(6\) with \(21000\) for \(6\) months \(\Rightarrow 21000\cdot 6=126000.\) 
So the ratio \(A:B:C=147000:189000:126000=7:9:6.\) One share \(=\dfrac{26400}{7+9+6}=\dfrac{26400}{22}=1200.\) 
Thus \(B=9\times1200=10800,\ C=6\times1200=7200.\) Excess \(=10800-7200=\boxed{3600}.\) 

Answer 2

Instead of comparing raw capital-month products, compute each partner's average effective capital over the full year by dividing their total capital-months by \( 12 \), then compare these average capitals directly.

\[ \bar{A}=\frac{16000\times3+11000\times9}{12}=\frac{147000}{12}=₹12{,}250,\quad \bar{B}=\frac{12000\times3+17000\times9}{12}=\frac{189000}{12}=₹15{,}750,\quad \bar{C}=\frac{21000\times6}{12}=₹10{,}500. \]

The profit ratio equals the ratio of these average capitals: \[ 12250:15750:10500 = 7:9:6 \quad(\text{dividing throughout by }1750). \]

1. Option A (\( 3600 \)): Each share unit equals \( \dfrac{26400}{7+9+6}=\dfrac{26400}{22}=1200 \). So \( B=9\times1200=₹10{,}800 \) and \( C=6\times1200=₹7{,}200 \), giving \( B-C=10800-7200=₹3600 \). This matches.
2. Option B (\( 2100 \)): This does not match the computed excess of \( ₹3600 \); rejected.
3. Option C (\( 3000 \)): This also does not match \( ₹3600 \); rejected.
4. Option D (\( 2300 \)): This is inconsistent with the computed value as well; rejected.

Using average effective capital confirms the profit-sharing ratio \( 7:9:6 \), and B's share exceeds C's by \( ₹3600 \).

Hence, the correct answer is 3600.