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P and Q started a business in partnership by investing in the ratio 7 ∶ 8. After 2 months of starting, P invested Rs. 20000 more, but withdrew Rs. 30000 after another 2 months. While, Q invested Rs. 40000 more after 3 months of starting, but withdrew Rs. 41000 after another 3 months. If at the end of the year, they divided the profit in the ratio 1 ∶ 3 then, with how much money did P and Q started the business?

P and Q started a business in partnership by investing in the ratio 7 ∶ 8. After 2 months of starting, P invested Rs. 20000 more, but withdrew Rs. 30000 after another 2 months. While, Q invested Rs. 40000 more after 3 months of starting, but withdrew Rs. 41000 after another 3 months. If at the end of the year, they divided the profit in the ratio 1 ∶ 3 then, with how much money did P and Q started the business? Correct Answer Rs. 22500

Let P and Q started the business by investing Rs. ‘7x’ and Rs. ‘8x’ respectively

Now, P invested Rs. 20000 more after 2 months of starting and withdrew Rs. 30000 after another 2 months so,

⇒ P’s investment for first 2 months = Rs. 7x

⇒ P’s investment for next 2 months = Rs. (7x + 20000)

⇒ P’s investment for remaining time 8 (= 12 - 2 - 2) months = (7x + 20000 - 30000) = Rs. (7x - 10000)

⇒ P’s total capital investment = (2 × 7x) + {2 × (7x + 20000)} + = 14x + 14x + 40000 + 56x - 80000 = 84x - 40000

Similarly,Q invested Rs. 40000 more after 3 months of starting and withdrew Rs. 41000 after another 3 months

⇒ Q’s investment for first 3 months = Rs. 8x

⇒ Q’s investment for next 3 months = Rs. (8x + 40000)

⇒ Q’s investment for remaining time 6 (= 12 - 3 - 3) months = (8x + 40000 - 41000) = Rs. (8x - 1000)

⇒ Q’s total capital investment = (3 × 8x) + {3 × (8x + 40000)} + {6 × (8x - 1000)} = 24x + 24x + 120000 + 48x - 6000 = 96x + 114000

Now,

⇒ Ratio of profit shares = Ratio of total capital investments

⇒ (84x - 40000)/(96x + 114000) = 1/3

⇒ 3 × (84x - 40000) = 96x + 114000

⇒ 3 × (21x - 10000) = 24x + 28500

⇒ 39x = 58500

⇒ x = 58500/39 = 1500

⇒ Total investment of P and Q when they started the business = 7x + 8x = 15x = 15 × 1500 = Rs. 22500

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