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**Hint:**Assume the time of investment for either person C or person A as some variable and find the time for the rest of the two. You can use the time of investment to find the ratio of shares of each person. Then change that ratio into a fraction to find the difference in shares.

**Complete step-by-step answer:**

Let’s make the question easier to understand first. Since all the time is dependent upon the number of months C invested the money, we can assume that number of months that C invested the money be $'m'$

Therefore, we can say from the question that

The person C invested $Rs.5600$ for $m$ months

The person B invested $Rs.4000$ for $\left( {m + 1} \right)$ months

And the person A invested $Rs.5500$ for $\left( {m + 1 + 2} \right) = \left( {m + 3} \right)$ months

Since we know what amount was invested and for how long, we can easily write the ratio of shares as the product of money and time for person A, B and C as:

$ \Rightarrow 5500 \times \left( {m + 3} \right):4000 \times \left( {m + 1} \right):5600 \times m$

$ \Rightarrow 55\left( {m + 3} \right):40\left( {m + 1} \right):56m$

Now, according to the question we can use these ratios of shares to find the difference of the shares of C and B and then equate them to $Rs.250$. Also, the total profit was given $Rs.6000$

Using the above relation, the difference in the shares of person C and B can be represented by:

$ \Rightarrow \dfrac{{56m - 40\left( {m + 1} \right)}}{{56m + 40\left( {m + 1} \right) + 55\left( {m + 3} \right)}}$

Therefore, according to the case given in the question:

$ \Rightarrow \dfrac{{56m - 40\left( {m + 1} \right)}}{{56m + 40\left( {m + 1} \right) + 55\left( {m + 3} \right)}} \times 6000 = 250$

Now let’s solve this equation for the value of $'m'$

$ \Rightarrow \dfrac{{56m - 40m - 40}}{{56m + 40m + 40 + 55m + 165}} \times 600 = 25$

$ \Rightarrow \dfrac{{16m - 40}}{{151m + 205}} \times 120 = 5$

$ \Rightarrow \left( {16m - 40} \right) \times 24 = 151m + 205$

\[ \Rightarrow 384m - 960 = 151m + 205 \Rightarrow 233m = 1165\]

$ \Rightarrow m = 5$

We got the value of $'m'$, which was our first assumption for the number of months of investments for person C.

Also, for person A we have $\left( {m + 1 + 2} \right) = \left( {m + 3} \right) = 5 + 3 = 8$ months

Hence, the time for which person A invested the money was $8$ months.

**Note:**Try to be careful with the signs and brackets while solving for the ratios. Write all the given information for making things less complicated. An alternative approach can be that we assume the number of months for person A as $'m'$. Then after the calculation of the difference equation, we get our answer.

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