Question

M and N start a business with respective capitals of Rs. 35,000 and Rs. 22,000. M withdrew an amount of Rs. 1,000 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 N?

• A. Rs. 22,500
• B. Rs. 33,000
• C. Rs. 43,500
• D. Rs. 46,000
• E. Rs. 48,500

Hint:

Add up each partner's capital for all 13 months (an arithmetic progression, since Rs. 1,000 changes hands every month) to get the profit-sharing ratio, then split Rs. 85,500 in that ratio.

Answer 1

The Correct Option is C

Step 1: Understand how the capitals change.
Since M withdraws Rs. 1,000 every single month for all 13 months, and N deposits an extra Rs. 1,000 every single month for all 13 months, by the time the business closes M has withdrawn a total of \(13\times1000=13{,}000\) and N has added a total of \(13\times1000=13{,}000\). So M's capital during month \(k\) (for \(k=1,2,\dots,13\)) is \(35000-1000k\), and N's capital during month \(k\) is \(22000+1000k\).

Step 2: List the capital for each month.
M: 34000, 33000, 32000, ..., down to \(35000-13000=22000\) in month 13.
N: 23000, 24000, 25000, ..., up to \(22000+13000=35000\) in month 13.

Step 3: Add up the capital-months (the profit-sharing weight) for each partner.
M's series is an AP with first term 34000, last term 22000, 13 terms:
$$\text{Sum}_M=13\times\frac{34000+22000}{2}=13\times28000=364000$$
N's series is an AP with first term 23000, last term 35000, 13 terms:
$$\text{Sum}_N=13\times\frac{23000+35000}{2}=13\times29000=377000$$

Step 4: Find the profit-sharing ratio.
$$M:N=364000:377000=364:377$$
Both numbers are divisible by 13: \(364/13=28\) and \(377/13=29\). So \(M:N=28:29\), and the total of the ratio parts is \(28+29=57\).

Step 5: Calculate N's share of the profit.
$$N\text{'s share}=85500\times\frac{29}{57}=1500\times29=43{,}500$$
(Since \(85500\div57=1500\) exactly.)
So N's share of the profit is Rs. 43,500. The correct option is (c) Rs. 43,500.