A certain sum of money trebles in 6 years by some rate of simple interest. In…

For simple interest, the interest earned each year is constant (SI = P x r x t / 100), so it grows in direct proportion to time. This means the interest needed to make a sum grow from the principal to any given multiple of itself is also directly proportional to the time taken -- equal multiples of the principal earned as interest always take equal amounts of time, regardless of which multiple is being reached.

1. A sum trebles (becomes 3 times itself) in 6 years, so the interest earned in these 6 years equals 3P minus P, that is 2P (twice the principal).

2. To become five times itself, the interest needed is 5P minus P, that is 4P -- twice as much interest as was earned in the first 6 years.

3. Since simple interest accrues at a constant rate per year, earning twice as much interest takes twice as much time: 6 x 2 = 12 years.

Check using the rate: 2P = P x r x 6 / 100 gives r = 100/3 percent per annum. Over 12 years, interest = P x (100/3) x 12 / 100 = 4P, so the amount = P + 4P = 5P -- confirming the sum becomes five times itself in 12 years.

Hence, the sum takes 12 years to become five times itself.