An amount is said to double in 5 years with compound interest. How many years will it take for the amount to grow to 8 times its original value?

Correct — A. Under compound interest the amount is multiplied by the same factor in every equal block of time. Here the factor is ×2 per 5 years.

After 5 years: P × 2 = 2P
After 10 years: 2P × 2 = 4P
After 15 years: 4P × 2 = 8P

8 = 2³, so three doublings: 3 × 5 = 15 years → option (a).

Why the others are wrong

• (b)
  16 — At 15 years the sum is already 8 times. 16 years is 3.2 doubling periods, which makes it about 9.2 times the principal — past the target.
• (c)
  17 — 17 years is 3.4 doubling periods, about 10.6 times the principal. The target of 8 times is reached at exactly three doublings, two years earlier.
• (d)
  18 — 18 years is 3.6 doubling periods, about 12.1 times the principal. 18 is not a multiple of the 5-year doubling time, so no whole number of doublings lands on it.

Compound interest grows by multiplication, not addition. If P becomes 2P in 5 years, then (1 + r)⁵ = 2, and every later 5-year block multiplies the amount by 2 again.

So the amount after n blocks is P × 2ⁿ. A target that is a power of the growth factor — 4, 8, 16 times — is reached in a whole number of blocks.

The annual rate is never needed. Working it out (about 14.9% a year) only adds rounding error.

Key facts

• Under compound interest, if a sum becomes k times in T years, it becomes kⁿ times in nT years.
• 8 = 2³, so a sum that doubles in 5 years becomes 8 times in 3 × 5 = 15 years.
• Under simple interest the interest added each year is fixed, so a sum that doubles in 5 years needs 35 years to become 8 times.

Common traps

• Treating the growth as simple interest and adding one principal every 5 years, which gives 35 years.
• Solving for the annual rate first and compounding year by year, which is slow and invites rounding error.

No directly related past PYQ was found.