If you invest $1000 at 8% compounded rate, how long wil it take to grow to $4000?

Question

BlackTom AI · Accepted Answer

Compound interest follows A = P(1 + r)^t, so use A = 4000, P = 1000, r = 0.08 to solve for t.

1) Relevant concepts:
- Compound interest multiplies the principal by (1 + r) each compounding period (here yearly).
- To find time when amount reaches a given multiple of the principal, solve an exponential equation using logarithms.
- A quick check is the Rule of 72: doubling time ≈ 72/r (in percent), so at 8% a doubling ≈ 9 years; quadrupling = two doublings ≈ 18 years.

2) Step-by-step reasoning and calculations:
Start with the compound formula:
4000 = 1000(1.08)^t
Divide both sides by 1000:
4 = (1.08)^t
Take natural logarithms of both sides:
ln(4) = t · ln(1.08)
Solve for t:
t = ln(4) / ln(1.08)

3) Intermediate numerical steps:
ln(4) ≈ 1.38629436112
ln(1.08) ≈ 0.07696104113
t ≈ 1.38629436112 / 0.07696104113 ≈ 18.01 years

4) Why the correct answer is correct:
Because (1.08)^18 ≈ 4, eighteen years of 8% annual compounding multiplies the initial $1000 by about 4, producing $4000. The logarithmic solution exactly finds the time required for the exponential growth factor to reach 4.

5) Why the other options are incorrect:
- 9 years: This is the approximate doubling time at 8% (Rule of 72), not the time to quadruple; one doubling (9 years) gives $2000, not $4000.
- 12.5 years: No standard compound or simple-interest calculation yields 12.5 for quadrupling at 8%; it understates the needed time.
- 25 years: Overestimates the required time; at 8% you reach quadruple earlier (≈18 years).

6) Conclusion:
The correct time to grow $1000 to $4000 at 8% compounded annually is approximately 18 years, matching the provided answer "18 years."

If you invest $1000 at 8% compounded rate, how long wil it take to grow to $4000?单项选择题

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