A certain sum becomes 4 times itself in 8 years at simple interest. In how many years will it become 12 times itself at the same rate?

Correct — A. At simple interest the interest grows in step with time, so track the interest, not the amount.

4 times itself → interest = 4P − P = 3P in 8 years
Interest per year = 3P ÷ 8 = 3P⁄8 (a 37.5% rate)

12 times itself → interest needed = 12P − P = 11P
Time = 11P ÷ 3P⁄8 = 11 × 8⁄3 = 88⁄3 years
88⁄3 = 29 1⁄3 years → option (a)

Why the others are wrong

• (b)
  30 1⁄6 years — 30 1⁄6 years at 37.5% a year earns 11.3125P of interest, so the sum would reach 12.3125 times itself, a little past 12.
• (c)
  28 2⁄5 years — 28 2⁄5 years at 37.5% a year earns 10.65P, so the sum reaches only 11.65 times itself, short of 12.
• (d)
  27 3⁄7 years — 27 3⁄7 years at 37.5% a year earns 72⁄7 ≈ 10.29P, so the sum reaches about 11.29 times itself, short of 12.

Under simple interest each year adds the same fixed slice of the principal, so interest is proportional to time.

"Becomes n times itself" means the interest is (n − 1) times the sum: 4 times means 3P of interest, 12 times means 11P.

So time scales with n − 1, not with n: 8 × 11⁄3 = 88⁄3 years.

Key facts

• At simple interest, becoming n times itself means interest = (n − 1) × principal.
• Simple interest at a fixed rate is proportional to time.
• Here the rate is 3⁄8 of the principal a year, which is 37.5%.

Common traps

• Scaling the amount instead of the interest. 4 times in 8 years does not make 12 times in 24 years: 24 years earns 9P, so the sum is only 10 times itself.
• Taking 12P as the interest. That gives 12 × 8⁄3 = 32 years, when the interest needed is 11P.

No directly related past PYQ was found.