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 the value of x is = ?

Explanation:-

Clearly, A invested his capital for 12 months
while each one of B and C invested his capital for (12 - x) months
Ratio of profits os A, B, C
$$= \left( {50000 \times 12} \right)$$   : $$\left[ {60000 \times \left( {12 - x} \right)} \right]$$     : $$\left[ {70000 \times \left( {12 - x} \right)} \right]$$
$$\eqalign{ & = 60:6\left( {12 - x} \right):7\left( {12 - x} \right) \cr & {\text{But ratio of profits}} \cr & = 20:18:21 \cr & = 60:54:63 \cr}$$
$$\therefore 60:\left( {72 - 6x} \right):\left( {84 - 7x} \right)$$     = $$60$$ : $$54$$ : $$63$$
$$\eqalign{ & {\text{So, }}72 - 6x = 54 \cr & \Rightarrow 6x = 18 \cr & \Rightarrow x = 3 \cr}$$