A family paid $99,000 cash for a house. Fifteen years later, the house was sold for $195,000.

If interest is compounded continuously, what annual nominal rate of interest did the original

$99,000 investment earn?

To determine the annual nominal interest rate earned by the original investment, we'll use the formula for continuous compounding:

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

where:

- $ A $ = future value (final amount)
- $ P $ = initial principal (starting amount)
- $ r $ = annual nominal interest rate
- $ t $ = time in years
- $ e $ = Euler's number, approximately 2.71828

### Step 1: Assign Known Values

- $ A = 195,000 $ (the amount received after 15 years)
- $ P = 99,000 $ (the initial amount)
- $ t = 15 $ (years)

We want to solve for $ r $, the interest rate.

### Step 2: Set Up the Formula

\[
195,000 = 99,000 \cdot e^{15r}
\]

### Step 3: Solve for $ r $

1. Divide both sides by $ 99,000 $:

   \[
   \frac{195,000}{99,000} = e^{15r}
   \]

   \[
   1.9697 = e^{15r}
   \]

2. Take the natural logarithm ($ \ln $) of both sides to isolate $ r $:

   \[
   \ln(1.9697) = 15r
   \]

3. Calculate $ \ln(1.9697) $:

   \[
   \ln(1.9697) \approx 0.676
   \]

4. Divide by 15:

   \[
   r \approx \frac{0.676}{15}
   \]

   \[
   r \approx 0.0451
   \]

### Step 4: Convert to Percentage

\[
r \approx 0.0451 \times 100\% = 4.51\%
\]

### Final Answer:

The annual nominal interest rate that the original \$99,000 investment earned, when compounded continuously, is approximately **4.51%**.

Would you like to see the steps in more detail, or do you have any questions?

Here are 5 related questions:
1. What would be the formula if the interest was compounded annually instead of continuously?
2. How much would the investment be worth if it was compounded continuously at a 4.51% interest rate for 20 years?
3. What is the difference between nominal and effective interest rates?
4. How does the continuous compounding formula compare to the formula for compound interest applied a finite number of times per year?
5. If the house was sold for a different amount, say \$250,000, how would that change the nominal interest rate?

**Tip:** When using continuous compounding, the formula $ A = P \cdot e^{rt} $ helps because it simplifies calculations for growth over time, especially when changes are applied continuously rather than at discrete intervals.

A family paid $99,000 cash for a house. Fifteen years later, the house was sold for $195,000. If interest is compounded continuously, what annual nominal rate of interest did the original $99,000 investment earn?

Learn how to calculate the continuous compound interest rate for a $99,000 investment over 15 years that grows to $195,000. This problem covers key mathematical concepts, including the continuous compounding formula, logarithmic calculations, and exponential growth. Suitable for students in Grades 10-12.

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