Answer: Option B
Let's assume that the profit earned by the business in one year is x. Then, the profit earned by A and B in one year will be:
A's profit = (1/2)3000x + 6*(1/2)3000(x/2) = 2250xB's profit = 4000x + 63000*(x/2) = 19000x/6 = 3167x
Now, let's find the ratio in which A and B divide the profit.
Let A's share of profit be 9k and B's share be 8k. Then, we have:
9k + 8k = 2250x + 3167x=> 17k = 5417x=> k = 5417x/17
Therefore, A's share of profit = 9*(5417x/17) = 2883xAnd, B's share of profit = 8*(5417x/17) = 2534x
Hence, the ratio in which A and B divide the profit is 2883x : 2534x, which simplifies to 9 : 8.
Therefore, the correct option is B) 9 : 8.
Some important formulas used in the above solution are:

• Profit = Total earnings - Total expenses
• Ratio = (part 1 / total) : (part 2 / total)
• A's profit = (1/2)A's capital + 6(1/2)A's capital(A's capital/initial capital)
• B's profit = B's capital + 6*(A's capital/initial capital)*B's capital

Where A and B are the amounts invested by A and B respectively, and x is the total profit earned by the business in one year.
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