[Varun Mohata]

To solve this problem, we need to determine the share of A in the partnership’s annual gain of Rs. 18,000.

Let’s assume that A invests x amount of money at the beginning. Then, after 6 months, B invests double the amount, so B’s investment is 2x. After 8 months, C invests thrice the amount, so C’s investment is 3x.

The share of each partner is generally determined by the ratio of their investments multiplied by the time for which they invested.

Now, let’s calculate the shares of A, B, and C:

A’s share = (A’s investment × A’s time) / (Total investment × Total time) = (x × 12) / (6x + 2x + 3x) [Total investment = A’s investment + B’s investment + C’s investment, Total time = 12 months]

Simplifying the expression, we get:

A’s share = 12x / 11x = 12 / 11

To find the value of A’s share in terms of the annual gain, we multiply A’s share by the total annual gain:

A’s share = (12 / 11) × 18,000 = 32,727.27 (rounded to two decimal places)

Therefore, A’s share in the partnership’s annual gain of Rs. 18,000 is approximately Rs. 32,727.27.

So. A’s Share = Rs 32727.

[Varun Mohata]

Let’s start by calculating the interest earned by Rs. 4,000. We can use the formula for simple interest:

Interest = Principal × Rate × Time

Given that the principal (P) is Rs. 4,000 and the rate (R) is 3% per annum, we need to determine the time (T) in years.

Now, let’s calculate the interest earned by Rs. 4,000 using the formula:

Interest = (4000 × 3 × T) / 100

Simplifying the expression, we get:

Interest = (120T) / 100 Interest = (6T) / 5

Next, let’s calculate the interest earned by Rs. 5,000 over 5 years at a rate of 4% per annum:

Interest = (5000 × 4 × 5) / 100 Interest = 1000

So, we have the equation: (6T) / 5 = 1000

To solve for T, we can cross-multiply and solve for T:

6T = 5 × 1000 6T = 5000 T = 5000 / 6 T ≈ 833.33 years

Therefore, it will take approximately 833.33 years for Rs. 4,000 lent at 3% per annum to earn as much interest as Rs. 5,000 will earn in 5 years at 4% per annum.