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**Hint**: The given questions revolve around the concepts of partnership and proportions. So, one must have an idea of how partnerships work, how investments are made and profits are distributed. We are required to find the investment made by C if all the three partners got an equal share of profit. We are also given the investment made by A. There are mainly two types of relations between any two given variables: direct relation or direct proportion and inverse relation or inverse proportion.

**:**

__Complete step-by-step answer__In the given problem, A started a business and invested an amount of $Rs10,000$ initially.

So, investment of A $ = Rs10,000$

Time duration of investment of A $ = 12months$

Also, B joined A four months later.

So, Time duration of investment of B $ = 8months$

Similarly, C joined in the partnership two more months later.

So, Time duration of investment of B $ = 6months$

Now, we know that to have an equal share of profits, the investment made has to be in inverse proportion to the time duration for which investment is made.

So, for equal profits, $Investment \propto \dfrac{1}{{Time}}$.

So, making a table for the investment and time duration of investment for A and C, we get,

Person | Investment | Time duration |

A | $Rs\,10,000$ | $12$ months |

C | $Rs\,x$ | $6$ months |

Now, calculating the value of x following the inverse relation between investment and time duration of investment, we get,

$ \Rightarrow x = \dfrac{{10,000 \times 12}}{6}$

Cancelling the common factors in numerator and denominator, we get,

$ \Rightarrow x = Rs\,20,000$

So, the investment of C is $Rs\,20,000$.

Hence, option (B) is correct.

**So, the correct answer is “Option B”.**

**Note**: In the question, we need to find out the relation between the different terms given to us. We must know how to find missing variables and constants in direct and inverse proportions in order to solve such questions. One must take care while doing the calculations so as to be sure of the answer.

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