A certain sum of money becomes seven times itself when invested at a certain rate of simple interest, in 14 years. How much time (in years, rounded off to 2 decimal places) will it take to become 18 times itself at the same rate?

Correct — D. Under simple interest the yearly interest never changes, so the interest grows in proportion to time — the total does not.

Becoming 7 times itself means interest of 6P, earned in 14 years.
So 1P of interest takes 14 ÷ 6 years.

Becoming 18 times itself means interest of 17P.
Time = 17 × 14 ÷ 6 = 238 ÷ 6 = 39.666…

Rounded to two decimal places: 39.67 years — option (d).

Why the others are wrong

• (a)
  35.5 — At 35.5 years the interest earned is 35.5 × 6 ÷ 14 = 15.21P, taking the sum to 16.21 times itself — short of 18.
• (b)
  27.33 — 27.33 years earns 27.33 × 6 ÷ 14 = 11.71P, so the sum reaches only 12.71 times itself.
• (c)
  42.78 — 42.78 years earns 42.78 × 6 ÷ 14 = 18.33P, taking the sum to 19.33 times itself — past the target.

Simple interest pays the same amount every year, so time is proportional to the interest, not to the multiple the sum reaches.

Strip the principal out: a sum that becomes n times itself has earned (n − 1) times the principal. Seven times in 14 years means 6P in 14 years.

So the time for 17P is 17/6 of 14 years. The rate never needs computing, though it is available: 6P = P × r × 14 ÷ 100 gives r = 600/14, about 42.86% per annum.

Key facts

• Under simple interest, the time to reach n times the principal is proportional to (n − 1).
• Becoming 7 times in 14 years fixes the rate at 600/14, about 42.86% per annum.
• 238 ÷ 6 = 39.666…, which is 39.67 to two decimal places.

Common traps

• Scaling by the multiple itself: 14 × 18 ÷ 7 = 36 years, which forgets that only the interest grows
• Using 18 units of interest instead of 17
• Truncating 39.666… to 39.66 instead of rounding it

No directly related past PYQ was found.