How much money invested at 6% compounded continuously for 5 years will result in $916?
The Correct answer and Explanation is:
To find out how much money must be invested at a continuous compounding interest rate of 6% for 5 years to result in $916, we can use the formula for continuous compounding, which is given by:A=PertA = Pe^{rt}A=Pert
Where:
- AAA is the amount of money accumulated after time ttt,
- PPP is the principal amount (the initial amount of money),
- rrr is the annual interest rate (decimal),
- ttt is the time the money is invested for (in years),
- eee is the base of the natural logarithm, approximately equal to 2.71828.
Given:
- A=916A = 916A=916
- r=0.06r = 0.06r=0.06 (6% as a decimal)
- t=5t = 5t=5
We can rearrange the formula to solve for PPP:P=AertP = \frac{A}{e^{rt}}P=ertA
Substituting the known values into the equation:P=916e0.06×5P = \frac{916}{e^{0.06 \times 5}}P=e0.06×5916
First, we calculate rtrtrt:0.06×5=0.300.06 \times 5 = 0.300.06×5=0.30
Now calculate e0.30e^{0.30}e0.30:e0.30≈1.34986e^{0.30} \approx 1.34986e0.30≈1.34986
Now, substitute back into the equation for PPP:P=9161.34986≈678.70P = \frac{916}{1.34986} \approx 678.70P=1.34986916≈678.70
Therefore, approximately $678.70 must be invested.
Explanation
Continuous compounding is a concept where interest is calculated and added to the principal amount at every possible instant, rather than at specific intervals (like annually or monthly). The formula for continuous compounding highlights how the amount of money grows exponentially over time due to the nature of compounding.
In this scenario, we want to determine how much principal investment is necessary to achieve a future value of $916 after 5 years at a continuous compounding rate of 6%. By rearranging the standard formula, we isolate the principal PPP, allowing us to plug in the known values of the future amount AAA, the interest rate rrr, and the time period ttt.
After performing the calculations, we find that an initial investment of approximately $678.70 will grow to $916 after 5 years at a continuous rate of 6%. This demonstrates the power of compounding, particularly when interest is compounded continuously, allowing investments to grow at a faster rate compared to discrete compounding intervals.