In a joint business, the capitals of A and B are in the ratio $3:4$ and the capitals of B and C are in the ratio $6:5$. If A receives a profit share of $\text{Rs. }450$, find the profit share of C.

We are given two separate capital ratios:
$$\text{A}:\text{B}=3:4$$

$$\text{B}:\text{C}=6:5$$

To form a continuous ratio $\text{A}:\text{B}:\text{C}$, we must equalize the term corresponding to B. The LCM of $4$ and $6$ is $12$:

$$\text{A}:\text{B}=(3\times 3):(4\times 3)=9:12$$

$$\text{B}:\text{C}=(6\times 2):(5\times 2)=12:10$$

Combining these gives the absolute capital ratio:

$$\text{A}:\text{B}:\text{C}=9:12:10$$

Profits are split in this identical ratio. Let the corresponding values be $9k$, $12k$, and $10k$.
Given A's profit share = $\text{Rs. }450$:

$$9k=450⟹k=50$$

Thus, the profit share of C is:

$$\text{Profit of C}=10k=10\times 50=\text{Rs. }500$$