A started a business with ₹30,000. B joined after x months with ₹40,000. At the end of the year, their profit-sharing ratio is 3:2. What is the value of x?
6 months
In a partnership, each partner's share of profit is proportional to the product of the capital invested and the time for which it is invested.
A invests ₹30,000 for the full 12 months, so A's capital-months = \(30000 \times 12 = 360000\).
B joins after x months, so B invests ₹40,000 for \((12 - x)\) months, giving capital-months = \(40000 \times (12 - x)\).
The profit-sharing ratio A : B is given as 3 : 2, so \(\dfrac{30000 \times 12}{40000 \times (12 - x)} = \dfrac{3}{2}\).
Simplify the left side: \(\dfrac{360000}{40000(12 - x)} = \dfrac{9}{12 - x}\).
So \(\dfrac{9}{12 - x} = \dfrac{3}{2}\), which gives \(3(12 - x) = 18\) by cross-multiplication.
Then \(12 - x = 6\), so \(x = 6\).
Hence B joined after 6 months.
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