Question

M and N start a business with respective capitals of Rs. 35,000 and Rs. 22,000. M withdrew an amount of Rs. 1000 every month from the business while N put in an additional amount of Rs. 1,000 every month into the business. If they close the business after 13 months after making a profit of Rs. 85,500, then what is the share of M?

• A. Rs. 22,000
• B. Rs. 33,000
• C. Rs. 42,000
• D. Rs. 46,000

Hint:

Add up each partner's capital for every one of the 13 months to get their capital-month totals, then split the profit in that ratio.

Answer 1

The Correct Option is C

Step 1: Write M's capital for each of the 13 months.
M withdraws Rs. 1000 every month, so in month k (k = 1 to 13), M's capital is \[ 35000-1000k \] This gives 34000 in month 1, decreasing steadily down to 22000 in month 13.

Step 2: Sum M's capital-months.
\[ \sum_{k=1}^{13}(35000-1000k) = 13\times35000-1000\sum_{k=1}^{13}k \] Since \(\sum_{k=1}^{13}k = \dfrac{13\times14}{2}=91\), \[ = 455000-1000\times91 = 455000-91000 = 364000 \]

Step 3: Write and sum N's capital-months the same way.
N's capital in month k is \(22000+1000k\), so \[ \sum_{k=1}^{13}(22000+1000k) = 13\times22000+1000\times91 = 286000+91000 = 377000 \]

Step 4: Find the profit-sharing ratio.
\[ M:N = 364000:377000 \] Dividing both by 13000 (their common factor), \[ M:N = 28:29 \]

Step 5: Split the profit in this ratio.
Total parts \(=28+29=57\), and each part is worth \(85500/57=1500\).
\[ \text{M's share} = 28\times1500 = 42000 \]

Final Answer:
M's share of the profit is Rs. 42,000. \[ \boxed{Rs.\ 42{,}000} \]