aved At 6.2 percent interest, how long does it take to double your money? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) 10 points Length of time years eBook At 6.2 percent interest, how long does it take to quadruple it? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) Print Length of time References years
The Correct Answer and Explanation is:
To determine how long it takes to double or quadruple your money at a 6.2 percent interest rate compounded annually, we use the formula derived from the rule of exponential growth:
FV=PV⋅(1+r)tFV = PV \cdot (1 + r)^t
Where:
- FVFV is the future value
- PVPV is the present value
- rr is the annual interest rate in decimal form
- tt is the number of years
1. Time to Double
For doubling, FV/PV=2FV/PV = 2, so:
2=(1+0.062)t2 = (1 + 0.062)^t
Taking natural logarithms on both sides:
ln(2)=t⋅ln(1.062)\ln(2) = t \cdot \ln(1.062)
t=ln(2)ln(1.062)≈0.69310.06017≈11.52 yearst = \frac{\ln(2)}{\ln(1.062)} \approx \frac{0.6931}{0.06017} \approx 11.52 \text{ years}
2. Time to Quadruple
For quadrupling, FV/PV=4FV/PV = 4, so:
4=(1+0.062)t4 = (1 + 0.062)^t
ln(4)=t⋅ln(1.062)\ln(4) = t \cdot \ln(1.062)
t=ln(4)ln(1.062)≈1.38630.06017≈23.04 yearst = \frac{\ln(4)}{\ln(1.062)} \approx \frac{1.3863}{0.06017} \approx 23.04 \text{ years}
Explanation
This problem hinges on understanding compound interest and exponential growth. Instead of solving iteratively or guessing, logarithms allow us to extract the unknown time variable from the exponent. The annual interest rate of 6.2 percent means your investment grows by a fixed percentage each year, compounding over time. Using natural logarithms aligns with exponential equations and ensures accuracy.
This method assumes annual compounding, making it simpler than cases involving monthly or continuous compounding. The key insight is recognizing that doubling and quadrupling are not linear; quadrupling does not take twice as long as doubling. Because of the nature of exponential growth, it takes just over twice the time to quadruple your money as it does to double it.
