Ram, Bharat and Sham signed agreement of partnership in the ratio $\frac{7}{2}:\frac{4}{3}:\frac{6}{5}$ . After 4 months, Ram increases his share 50%. If the total profit at the end of one year be Rs. 16,200, then sham's share in the profit is:

The correct option is Rs 2,700.

Step 1: Simplify the initial ratio of investments

The initial ratio of investment for Ram, Bharat, and Sham is given as:

$\frac{7}{2}:\frac{4}{3}:\frac{6}{5}$

To convert this fractional ratio into whole numbers, we find the Least Common Multiple (LCM) of the denominators (2, 3, and 5):

LCM(2, 3, 5) = 30

Multiplying each fraction in the ratio by 30:

Ram = $\frac{7}{2}\times 30=105$

Bharat = $\frac{4}{3}\times 30=40$

Sham = $\frac{6}{5}\times 30=36$

So, the simplified initial ratio of their investments is 105 : 40 : 36.

Step 2: Calculate the effective investment for the whole year (12 months)

Let the initial monthly investment values be 105x, 40x, and 36x for Ram, Bharat, and Sham respectively.

For Ram:

- For the first 4 months, his investment is 105x.

- After 4 months, he increases his share by 50%:

New investment for remaining 8 months = $105x+(50\%\text{of}105x)=105x+52.5x=157.5x$

Total equivalent investment of Ram for 12 months:

$(105x\times 4)+(157.5x\times 8)=420x+1260x=1680x$

For Bharat:

Bharat's investment remains unchanged for the entire 12 months:

$40x\times 12=480x$

For Sham:

Sham's investment remains unchanged for the entire 12 months:

$36x\times 12=432x$

Step 3: Find the profit-sharing ratio

The ratio of their profits is equal to the ratio of their effective annual investments:

Profit Ratio (Ram : Bharat : Sham) = 1680x : 480x : 432x

Dividing each term by 48x to simplify:

Ram's share ratio = $\frac{1680}{48}=35$

Bharat's share ratio = $\frac{480}{48}=10$

Sham's share ratio = $\frac{432}{48}=9$

Thus, the final profit ratio is 35 : 10 : 9.

Step 4: Calculate Sham's share in the total profit

Total ratio parts = 35 + 10 + 9 = 54 parts

Total profit = Rs 16,200

Sham's share = $\frac{9}{54}\times 16200$

Simplifying the fraction $\frac{9}{54}=\frac{1}{6}$:

Sham's share = $\frac{1}{6}\times 16200=2700$

Therefore, Sham's share in the profit is Rs 2,700.