-    Clearly, Manish invested his capital for 12 months, Vibhor for 9 months and Vivek for 3 months.
  So, ratio of their capitals = (45000 x 12) : (60000 x 9) : (90000 x 3)
  = 540000 : 540000 : 270000 = 2:2:1.
  ∴ Manish’s share = Rs. [16500 x 2/5] = Rs. 6600;
  Vibhors share = Rs. [16500 x 2/5] = Rs. 6600;
  Vivek’s share = Rs. [16500 x 1/5] = Rs. 3300.

To find the share of Vivek in the profit, we need to calculate the ratio of the investment of each person and the time period for which they invested in the business. The ratio of profit sharing for Manish, Vibhor, and Vivek is calculated as follows:

Manish's investment = Rs. 45,000 (He invested for the entire year)

Vibhor's investment = Rs. 60,000 (He invested for 9 months)

Vivek's investment = Rs. 90,000 (He invested for 6 months)

To find the ratio of profit sharing, we can multiply the investment by the time period for each person:

Manish's ratio = 45,000 × 12 = 5,40,000

Vibhor's ratio = 60,000 × 9 = 5,40,000

Vivek's ratio = 90,000 × 6 = 5,40,000

The total ratio of profit sharing is:

Manish + Vibhor + Vivek = 5,40,000 + 5,40,000 + 5,40,000 = 16,20,000

The share of Vivek can be calculated by dividing his ratio by the total ratio and then multiplying the result by the total profit:

Vivek's share = (Vivek's ratio / Total ratio) × Total profit

Vivek's share = (5,40,000 / 16,20,000) × 16,500

Vivek's share = 5,500

Therefore, the share of Vivek in the profit is Rs. 3300 (Option A).

To summarize, the steps to find the share of Vivek in the profit are:

• Calculate the investment and time period for each person.
• Find the ratio of profit sharing for each person by multiplying their investment and time period.
• Calculate the total ratio of profit sharing by adding the ratios of all persons.
• Find the share of Vivek by dividing his ratio by the total ratio and multiplying the result by the total profit.

If you think the solution is wrong then please provide your own solution below in the comments section .

 -  Suppose A invested Rs. 14a for 10 months and B invested Rs. 15a for b months. Then,
14a x 10 / 15a x b = 7/6 ⇒ b = 840 / 105 = 8

Hence, B invested the money for 8 months.

To solve this problem, we need to use the concept of the ratio of investments and the ratio of profits.

Let the initial investments of A and B be 14x and 15x, respectively.

Let the time for which A invested be 10 months and the time for which B invested be y months.

Since the profits are in the ratio 7:6, we can assume that the total profit is 13x. Therefore, A's profit would be (7/13) times the total profit and B's profit would be (6/13) times the total profit.

According to the question, A invested for 10 months and B invested for y months. Therefore, the ratio of the time for which they invested their money is 10:y.

We can now use the formula: Profit = (Investment × Time × Rate of Profit)/100

We know that A's profit was (7/13) times the total profit, and B's profit was (6/13) times the total profit. Therefore, we can write the following equations:

(7/13) × 14x × 10 = (6/13) × 15x × y

Simplifying this equation, we get:

y = (140/90) = 28/18 = 14/9 months

Therefore, B invested his money for 14/9 months, which is approximately 1.56 months, or 1 month and 16 days.

Since none of the options match this value, we can round it up to the nearest option, which is 8 months (Option D).

Hence, the correct answer is Option D, i.e., B invested his money for 8 months.

If you think the solution is wrong then please provide your own solution below in the comments section .