A sum becomes 2.5 times of itself in 5 years under simple interest. In how many years will it become 5 times itself at the same rate?

Correct — B.
Amount after 5 years = 2.5P, so interest = 2.5P − P = 1.5P
Interest per year = 1.5P ÷ 5 = 0.3P (a 30% rate)

Amount 5P means interest = 5P − P = 4P
Time = 4P ÷ 0.3P = 40⁄3
= 13 1⁄3 years → option (b)

Why the others are wrong

• (a)
  12 2⁄3 years — 12 2⁄3 years earns 0.3P × 38⁄3 = 3.8P of interest, so the sum reaches 4.8P, just short of 5P.
• (c)
  14 2⁄3 years — 14 2⁄3 years earns 0.3P × 44⁄3 = 4.4P of interest, taking the sum to 5.4P, past five times.
• (d)
  11 1⁄3 years — 11 1⁄3 years earns 0.3P × 34⁄3 = 3.4P of interest, so the sum reaches only 4.4P.

Under simple interest the same amount of interest is added every year, so interest is proportional to time. What scales with time is the interest, not the amount.

'2.5 times of itself' means the interest is 1.5 times the sum. 'Five times itself' means the interest is four times the sum.

So t = 5 × 4⁄1.5 = 40⁄3 years.

Key facts

• Under simple interest, a sum that becomes n times itself has earned (n − 1) times itself as interest
• At a fixed rate, simple interest is proportional to time
• Here the rate is 1.5⁄5 = 30% per annum

Common traps

• Scaling the amounts instead of the interest: 2.5 times in 5 years does not make 5 times in 10 years
• Treating the whole 2.5P as interest instead of subtracting the principal first

No directly related past PYQ was found.