The ratio of investment by A to that by B in a business is 14 :15 and the ratio of their respective profits at the end of a year is 2 : 5. If A invested the money for 3 months, then for how much time (in months) B invested his money?

Sure, let's solve this step by step:

Step 1: Understand the problem The problem is about the ratio of investments and profits of two people, A and B. We know the ratio of their investments and their profits. We also know how long A invested his money. We need to find out how long B invested his money.

Step 2: Use the formula for profit The profit from an investment is proportional to the amount of money invested and the time for which it is invested. This can be written as: Profit = Investment * Time

Step 3: Apply the formula to A and B For A, the investment is 14x (where x is some constant) and the time is 3 months. So, the profit is 14x * 3 = 42x. For B, the investment is 15x and we don't know the time. Let's call it y months. So, the profit is 15x * y.

Step 4: Use the ratio of profits We know that the ratio of A's profit to B's profit is 2:5. This means that 42x/15xy = 2/5.

Step 5: Solve for y Cross-multiplying gives us 210x = 30xy. Dividing both sides by 30x gives us y = 7 months.

So, B invested his money for 7 months.