Step 1: 
Use the formula for continuous compounding: 

\[A = P e^ { rt} \]- \(A\) is the amount of money accumulated after n years, including interest. 
- \(P\) is the principal amount (the initial amount of money). 
- \(r\) is the annual interest rate (decimal). 
- \(t\) is the time the money is invested for in years. 
- \(e\) is the base of the natural logarithm, approximately equal to 2.71828.

Step 2:

Given: 
- \(A = 916\) dollars 
- \(r = 0.06\) 
- \(t = 5\) years

Step 3:

Rearrange the formula to solve for \(P\): 
\[P = \frac { A} { e^ { rt} } \]

Step 4:

Substitute the given values into the formula: 
\[P = \frac { 916} { e^ { 0.06 \times 5} } \]

Step 5:

Calculate the exponent: 
\[e^ { 0.3} \approx 1.34986\]

Step 6:

Divide the amount by the calculated exponent: 
\[P = \frac { 916} { 1.34986} \approx 678.61\]

Supplemental Knowledge:

Continuous compounding in finance refers to an interest-accrual method in which interest is computed and added directly onto principal over time instead of at discrete intervals. The formula for continuous compounding derives from exponential function; specifically:

\[A = P e^ { rt} \]

Where: 
- \(A\) is the amount of money accumulated after n years, including interest. 
- \(P\) is the principal amount (the initial amount of money). 
- \(r\) is the annual interest rate (decimal). 
- \(t\) is the time the money is invested for, in years. 
- \(e\) is Euler's number (approximately 2.71828).

To find out how much money needs to be invested initially (\(P\)) to reach a certain amount (\(A\)) after a given time period with continuous compounding, you can rearrange the formula:

\[P = \frac { A} { e^ { rt} } \]

Applied Knowledge:

Imagine you are planning for your child's college education and you want to ensure there will be enough funds when they start classes in five years time. By understanding continuous compounding, you can calculate how much to invest now at certain interest rates to reach this financial goal.

As well, knowing how different compounding methods affect investments allows you to make educated choices when saving for retirement or large purchases like a house.

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