Questions: Sara checks her retirement account and sees that she has a total of 131,377 after 11 years. If her account earns 6.8% that is compounded on a continuous basis, what was the amount of her original investment?

Solution Steps

To find the original investment amount when the interest is compounded continuously, we can use the formula for continuous compounding: \( A = Pe^{rt} \), where \( A \) is the amount after time \( t \), \( P \) is the principal amount (original investment), \( r \) is the annual interest rate, and \( t \) is the time in years. We need to solve for \( P \).

1. Rearrange the formula to solve for \( P \): \( P = \frac{A}{e^{rt}} \).
2. Substitute the given values: \( A = 131377 \), \( r = 0.068 \), and \( t = 11 \).
3. Calculate \( P \) using these values.

To find the original investment amount \( P \) when the interest is compounded continuously, we use the formula: \[ A = Pe^{rt} \] where:

• \( A \) is the amount after time \( t \),
• \( P \) is the principal amount (original investment),
• \( r \) is the annual interest rate,
• \( t \) is the time in years.

We need to solve for \( P \). Rearranging the formula gives: \[ P = \frac{A}{e^{rt}} \]

Substituting the known values into the equation:

• \( A = 131377 \)
• \( r = 0.068 \)
• \( t = 11 \)

We have: \[ P = \frac{131377}{e^{0.068 \times 11}} \]

Calculating the exponent: \[ e^{0.068 \times 11} = e^{0.748} \approx 2.113 \]

Now substituting back into the equation for \( P \): \[ P = \frac{131377}{2.113} \approx 62182.34 \]

Final Answer