At what least number of complete years will a sum of money invested at 40% annual compound interest become more than three times (trebled) its original value?

Introduction / Context:
This problem deals with determining how many years it takes for an investment to grow to more than three times its original amount under compound interest. The annual interest rate is quite high at 40%, so the amount grows rapidly each year. The key idea is that the amount after n years of compound interest is given by a power expression in terms of the annual growth factor. We are asked for the smallest whole number of years at which the amount first exceeds three times the principal.

Given Data / Assumptions:

• Principal (P) is some positive amount (exact value is not required).
• Rate of interest (R) = 40% per annum.
• Interest is compounded annually.
• We seek the least integer n such that the amount is greater than 3P.
• There are no deposits or withdrawals during the investment period.

Concept / Approach:
Under annual compounding, the amount after n years is:
A = P * (1 + R / 100)^nFor R = 40%, the yearly growth factor is 1 + 40 / 100 = 1.4. We want A to be more than 3P, so we need:
P * (1.4)^n > 3P(1.4)^n > 3We then test successive integer values of n to find the smallest n that satisfies this inequality.

Step-by-Step Solution:
Step 1: Compute (1.4)^1 = 1.4, which is less than 3, so 1 year is not enough.Step 2: Compute (1.4)^2 = 1.4 * 1.4 = 1.96, still less than 3, so 2 years are not sufficient.Step 3: Compute (1.4)^3 = 1.96 * 1.4 = 2.744, still less than 3, so 3 years are not sufficient either.Step 4: Compute (1.4)^4 = 2.744 * 1.4 ≈ 3.8416.Step 5: Now (1.4)^4 ≈ 3.84, which is clearly greater than 3.Step 6: Therefore, after 4 years the amount becomes more than three times the original principal.Step 7: Since 3 years is still less than triple and 4 years is more than triple, the least number of complete years required is 4.

Verification / Alternative check:
We can express the amounts in terms of P for extra clarity. After 1 year, A1 = 1.4P. After 2 years, A2 = 1.4 * 1.4P = 1.96P. After 3 years, A3 = 1.96 * 1.4P ≈ 2.744P. After 4 years, A4 = 2.744 * 1.4P ≈ 3.8416P. Only at year 4 does the value cross 3P. This confirms that the least integer number of years is 4.

Why Other Options Are Wrong:
Option 3 years gives A ≈ 2.744P, which is less than 3P, so it does not satisfy the condition that the money is more than trebled. Options 5, 6, and 7 years also satisfy the condition but they are not the least number of years, because the amount already exceeds 3P at year 4. Hence they do not answer the specific requirement of the problem. Only 4 years is both sufficient and minimal.

Common Pitfalls:
Some students may misinterpret "trebled" as simply adding 40% three times, which would give 120% total and only 2.2 times the original, not 3 times. Others may try to solve the inequality using approximate logarithms and then forget to round up to the next whole year. It is often simpler and safer in such questions to test small integer values step by step, especially when the time horizon is likely to be short.

Final Answer:
The least number of complete years required for the money to become more than three times its original value at 40% compound interest per annum is 4 years.