Shortcut Trick
Profit Ratio = (Capital1 × Time1) : (Capital2 × Time2)
Ratio of Capitals = 32000 : 20000 = 8 : 5.
Time for Vaibhav = x, Time for Ashish = (x − 6).
Ratio = (8 × x) : (5 × (x − 6)) = 12 : 5.
Solving 8x ÷ (5x − 30) = 12 ÷ 5 ⇒ 40x = 60x − 360 ⇒ 20x = 360 ⇒ x = 18.
∴ The correct answer is 18 months.
Alternate Method
Given: Vaibhav Investment = Rs. 32000, Ashish Investment = Rs. 20000, Profit Ratio = 12 : 5, Ashish joined after 6 months.
Formula: Profit = Investment × Time period
⇒ Let Vaibhav invested for x months.
⇒ Ashish joined 6 months later, so his time = (x − 6) months.
⇒ (32000 × x) ÷ (20000 × (x − 6)) = 12 ÷ 5
⇒ (32 × x) ÷ (20 × (x − 6)) = 12 ÷ 5
⇒ (8x) ÷ (5 × (x − 6)) = 12 ÷ 5
⇒ 8x ÷ (x − 6) = 12
⇒ 8x = 12x − 72
⇒ 4x = 72 ⇒ x = 18 months.
∴ The correct answer is 18 months.
Additional Information
Partnership Principle
In a partnership, the profit is distributed among partners in the ratio of the product of their investment and time duration.
Time Calculation
If a person joins 'n' months after the start, their investment duration is (Total Time − n).