Three partners shared the profit in a business in the ratio \[5:7:8\]. They had partnered for 14months, 8 months, and 7 months respectively. What was the ratio of their investments?
(a) $5:7:8$
(b) $20:49:64$
(c) $38:28:21$
(d) None of these

Hint: Assume that the investment of each of the partners is x, y, and z. Use the fact that the profit earned is directly proportional to the product of the amount invested and the time for which amount is invested. Write equations based on data given in the question. Simplify the equations to write all the variables in terms of variables ‘x’ and thus find the ratio between them.

Complete step by step answer:
We have data regarding the ratio of profit shared by each partner and the time for which each of them has partnered. We have to calculate the ratio of their investments.
Let’s assume that their investments were Rs.x, Rs.y, and Rs.z.
We know that the profit earned is directly proportional to the product of the amount invested and the time for which amount is invested.
As the first partner has partnered for 14 months, the profit of the first partner is $=Rs.x\times 14=14x$.
As the second partner has partnered for 8 months, the profit of the second partner is $=Rs.y\times 8=8y$.
As the third partner has partnered for 7 months, the profit of the third partner is $=Rs.z\times 7=7z$.
We know that the ratio of profit of first and second partner is \[5:7\]. Thus, we have $\dfrac{14x}{8y}=\dfrac{5}{7}$.
Simplifying the above equation, we have $\dfrac{7x}{4y}=\dfrac{5}{7}$.
Rewriting variable ‘y’ in terms of ‘x’ by rearranging the terms of the above equation, we have $y=\dfrac{49x}{20}.....\left( 1 \right)$.
We know that the ratio of profit of first and third partner is \[5:8\]. Thus, we have $\dfrac{14x}{7z}=\dfrac{5}{8}$.
Simplifying the above equation, we have $\dfrac{2x}{y}=\dfrac{5}{8}$.
Rewriting variable ‘y’ in terms of ‘x’ by rearranging the terms of the above equation, we have $y=\dfrac{16x}{5}.....\left( 2 \right)$.
We will now calculate the ratio between the amount invested by all the partners.
Using equation (1) and (2), we have $x:y:z=x:\dfrac{49x}{20}:\dfrac{16x}{5}$. As this is simply ratio, we can multiply it by 20.
Thus, we have $x:y:z=20x:49x:64x=20:49:64$.
Hence, the ratio between the investment of all the partners is $20:49:64$, which is option (b).

Note: One must keep in mind that it’s not necessary to find the exact amount invested by each of the partners. We can simply calculate the ratio between the amount invested based on the data given in the question. We can also solve this question by writing all the variables in terms of other variables such as ‘y’ or ‘z’.