If \[4\left( {A's{\rm{ capital}}} \right) = 6\left( {B's{\rm{ capital}}} \right) = 10\left( {C's{\rm{ capital}}} \right)\], then out of profit of \[Rs.7750\], what is C’s share?

Hint: Here, we need to find the C’s share. We will assume the capital of A, B and C to be any variable and then we will find the capital of each in terms of the assumed variable. Then we will find the ratio of the capital of A, B and C. Then we will find the C’s share using the ratio and the given profit and finally convert the ratio into a fraction.

Complete step-by-step answer:
Here, we need to find the C’s share.
Let \[4\left( {A's{\rm{ capital}}} \right) = 6\left( {B's{\rm{ capital}}} \right) = 10\left( {C's{\rm{ capital}}} \right) = k\]
On further simplification, we get
A’s capital \[ = \dfrac{k}{4}\]
B’s capital \[ = \dfrac{k}{6}\]
C’s capital \[ = \dfrac{k}{{10}}\]
Now, we will find the ratio between the \[A's\] capital, \[B's\] capital and the \[C's\] share.
Ratio \[ = \dfrac{k}{4}:\dfrac{k}{6}:\dfrac{k}{{10}}\] …………. \[\left( 1 \right)\]
We will find the factors of each number in the denominator. Therefore, we get
\[\begin{array}{l}4 = 2 \times 2\\6 = 2 \times 3\\10 = 2 \times 5\end{array}\]
Now, we will find the LCM of the denominators of them using the factor we obtained.
We can see that the LCM of 4, 6 and 10 is equal to \[2 \times 2 \times 3 \times 5 = 60\]
We will multiply 60 to each in equation \[\left( 1 \right)\].
Therefore, the ratio becomes;
\[ \Rightarrow \] Ratio \[ = 60 \times \dfrac{k}{4}:60 \times \dfrac{k}{6}:60 \times \dfrac{k}{{10}}\]
On further simplification, we get
\[ \Rightarrow \]Ratio \[ = 15:10:6\]
Now, we will find the value of C’s capital in fraction.
C’s capital \[ = \dfrac{6}{{15 + 10 + 6}} = \dfrac{6}{{31}}\]
It is given that the profit \[ = {\rm{Rs}}.7750\]
Now, we will find the C’s share which will be equal to the product of value of C’s capital in fraction and the total profit.
\[ \Rightarrow \] C’s share \[ = \dfrac{6}{{31}} \times 7750\]
On multiplying the terms, we get
\[ \Rightarrow \] C’s share \[ = {\rm{Rs}}.1500\]
This is the required value of C’s share.

Note: To find the share of C, we have to convert its ratio of capital into a fraction. We will divide the ratio of the capital of C by the sum of all ratios and then multiply it by the profit. Here, we are considering the fact that each person will get profit according to the amount they invested. That means the ratio of the capital of investment will be the same as the ratio of their share.