Let the investments of A, B, and C be represented as $A$, $B$, and $C$ respectively.

Step 1: Expressing the relationships

From the problem, we are given the following relationships:

• Thrice the investment of A is equal to twice the investment of B:
  $3A = 2B$
• Thrice the investment of A is also equal to four times the investment of C:
  $3A = 4C$

Step 2: Expressing investments in terms of A

Using the two equations, we can express the investments $B$ and $C$ in terms of $A$:

• From $3A = 2B$, solving for $B$, we get: $B = \frac{3A}{2}$
• From $3A = 4C$, solving for $C$, we get: $C = \frac{3A}{4}$

Step 3: Total profit sharing ratio

The ratio of the investments of A, B, and C will be the same as the ratio of their profit shares. Therefore, the profit-sharing ratio will be proportional to their investments. Hence, the ratio is:

$A : B : C = A : \frac{3A}{2} : \frac{3A}{4}$

To simplify this, we multiply through by 4 to eliminate the fractions:

$A : B : C = 4A : 6A : 3A$

So, the profit-sharing ratio is $4 : 6 : 3$.

Step 4: Total shares of profit

The total number of parts is $4 + 6 + 3 = 13$ parts.

Now, we know that C’s share of the total profit is Rs 4863. C’s share is $3$ parts out of the total 13 parts. Therefore:

$\frac{3}{13} \times \text{Total Profit} = 4863$

Solving for the total profit:

$\text{Total Profit} = \frac{4863 \times 13}{3} = 21021$

Step 5: Share of A in the profit

A’s share corresponds to 4 parts out of the total 13 parts. So, A’s share is:

$A's \, Share = \frac{4}{13} \times 21021 = 6468$

Thus, A's share in the profit is Rs 6468.

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Would you like further clarification on this solution?

Here are five related questions you might find interesting:

1. How do you calculate profit-sharing ratios from different investments?
2. If A, B, and C made different profits, how would the ratio change?
3. Can the ratio method be applied to more than three partners?
4. What happens if one partner withdraws or increases their investment midway?
5. How can profit-sharing affect decision-making in a partnership?

Tip: When solving ratio-based problems, always express all quantities in terms of one variable to simplify the solution.