A and B enter into a partnership with Rs.50000 and Rs. 60000 respectively. C joins them after 'x' months, contributing Rs.70000 and B leaves 'x' months before the end of the year. If they share the profit in the ratio of 20 : 18 : 21, then find the value of 'x'.

A invested his capital for 12 months 

given  that,
 B and C invested his capital for (12 - x) months

as we know that 

$$investment\times time = profit$$

Ratio of profits os A, B, C

=$$(50000\times 12): [60000\times (12−x)] :[70000\times (12−x)] =20:18:21$$

60:6(12−x):7(12−x)  =20:18:21

to equate multiply by 3 

20:18:21=60:54:63

so 

$$60:6\times (12−x):7\times (12−x) = 60:54:63$$

$$60:(72−6x):(84−7x) = 6060 : 5454 : 6363$$

So, 72−6x=54 

=6x=18

x=3