A sum triples in ten years under compound interest at a certain rate of interest, the interest is being compounded annually. In how many years, it would become nine times?

Correct — A, 20 years. Under annual compounding a sum multiplies by the same factor in every equal stretch of time. Ten years turns the principal into three times itself, so the ten-year growth factor is 3. Another ten years multiplies by 3 again, giving 9 times the principal after twenty years. Written out, if the annual factor is r then r¹⁰ = 3, and nine times means r^t = 9 = 3² = (r¹⁰)² = r²⁰, so t = 20. The rate itself is never needed. The general rule worth carrying is that under compound interest a sum that becomes x times in n years becomes x^k times in kn years — so trebling in ten years means nine times in twenty and twenty-seven times in thirty.

• (b)30 years — Thirty years gives 3³ = 27 times the principal, not 9. This is the answer for a question asking when the sum becomes twenty-seven times.
• (c)40 years — Forty years gives 3⁴ = 81 times. The option works only if a candidate multiplies the time by the multiple rather than working with powers.
• (d)50 years — Fifty years gives 3⁵ = 243 times. Far too long, and it comes from treating the growth as if it added rather than multiplied.

Compound interest is repeated multiplication, and every question about doubling, trebling or n-times growth is a question about powers. Simple interest behaves quite differently: there the increase is a fixed amount each year, so a sum that triples in ten years becomes five times in twenty and seven times in thirty, adding two principals every decade. Knowing which regime the question is in decides everything.

The whole item is settled by writing 9 as a power of 3. Once the multiple is expressed as a power of the multiple already given, the number of periods reads off directly. It is worth being suspicious of an answer that scales linearly here — thirty years for nine times looks natural because 9 is three times 3, and that is precisely the mistake the option list is built to catch. The reflex to build is: under compounding, multiples compose by multiplication, and the corresponding times add.

• Under compound interest, a sum that becomes x times in n years becomes x² times in 2n years and x³ times in 3n years.
• Trebling in ten years means nine times in twenty years and twenty-seven times in thirty.
• Formally, r¹⁰ = 3 and 9 = 3² = r²⁰, so twenty years is the answer.
• The rate of interest is never required for this class of question.
• Under simple interest the same sum would become five times in twenty years, not nine, since equal amounts are added each period.

• Scaling the time linearly with the multiple and answering 30 years.
• Applying simple-interest reasoning to a compound-interest stem.
• Trying to find the rate first, which is unnecessary and invites an arithmetic slip.

Asked as a powers-of-the-multiple item where the rate is deliberately withheld, so that the only route is through the growth factor.

• practice — not a real PYQ

  A sum doubles in 6 years under compound interest. In how many years will it become eight times?

  • (a)12 years
  • (b)18 years
  • (c)24 years
  • (d)48 years

  Answer(b) 18 years — 8 = 2³, so three doubling periods of 6 years each.

• practice — not a real PYQ

  A sum becomes four times in 12 years under simple interest. In how many years will it become seven times?

  • (a)18 years
  • (b)21 years
  • (c)24 years
  • (d)28 years

  Answer(c) 24 years — under simple interest three principals of interest accrue in 12 years, so one principal takes 4 years and six principals take 24.