Question

Anuj, Bheem and Charles invest in the ratio \(2:4:5\). After 6 months Anuj withdrew \(\frac{1}{4}\) of his capital. Bheem withdrew \(\frac{1}{4}\) of his initial capital at the end of every quarter, while Charles added \(\frac{2}{5}\) of his initial capital at the end of every quarter. If the annual profit is Rs. 98000, what is Bheem's share?

• A. Rs. 14000
• B. Rs. 18000
• C. Rs. 20000
• D. Rs. 21000

Hint:

Instead of summing capital-months quarter by quarter, try computing each partner's average capital over the year and multiplying by 12.

Answer 1

The Correct Option is C

Step 1: Take initial capitals as $2k, 4k, 5k$ for Anuj, Bheem and Charles.
Step 2: Anuj: $2k$ for the first 6 months, then $1.5k$ for the next 6 months (withdrew $\frac{1}{4}$ of $2k=0.5k$). Average capital over 12 months $=\dfrac{2k(6)+1.5k(6)}{12}=1.75k$, so capital-months $=1.75k\times 12=21k$.
Step 3: Bheem: withdraws $\frac{1}{4}$ of his INITIAL capital ($1k$) at the end of every quarter, giving quarterly capitals $4k,3k,2k,1k$. Average capital $=\dfrac{4k+3k+2k+1k}{4}=2.5k$, so capital-months $=2.5k\times 12=30k$.
Step 4: Charles: adds $\frac{2}{5}$ of his initial capital ($2k$) every quarter, giving quarterly capitals $5k,7k,9k,11k$. Average capital $=\dfrac{5k+7k+9k+11k}{4}=8k$, so capital-months $=8k\times 12=96k$.
Step 5: Total capital-months $=21k+30k+96k=147k$.
Step 6: Bheem's share of the annual profit $=\dfrac{30k}{147k}\times 98000=\dfrac{30}{147}\times 98000=20000$.
\[\boxed{\text{Rs. }20000}\]