Quincy earned some money doing odd jobs last summer and put it in a savings account that earns 14% interest compounded continuously. After 3 years, there is $1,000.00 in the account. How much did Quincy earn doing odd jobs?

Question

Use the formula A=Pert, where A is the balance (final amount), P is the principal (starting amount), e is the base of natural logarithms (≈2.71828), r is the interest rate expressed as a decimal, and t is the time in years.
Round your answer to the nearest cent.
$

Answer 1

To find out how much Quincy earned doing odd jobs (the principal amount \( P \)), we will use the formula for continuous compounding:

\[ A = Pe^{rt} \]

In this situation:

• \( A = 1000 \) (the amount after 3 years)
• \( r = 0.14 \) (14% interest rate expressed as a decimal)
• \( t = 3 \) (the time in years)

We need to solve for \( P \):

Rearranging the formula gives:

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

Now we can substitute the known values into the equation:

1. Calculate \( rt \):

\[ rt = 0.14 \times 3 = 0.42 \]

2. Calculate \( e^{rt} \):

Using the approximate value of the base of natural logarithms \( e \approx 2.71828 \):

\[ e^{0.42} \approx 2.71828^{0.42} \approx 1.519 \]

3. Substitute back to find \( P \):

\[ P = \frac{1000}{1.519} \]

Calculating this gives:

\[ P \approx \frac{1000}{1.519} \approx 657.51 \]

Thus, rounding to the nearest cent, the amount Quincy earned doing odd jobs is:

\[ \boxed{657.51} \]