On your tenth birthday, you received $100 which you invested at 4.5 percent interest, compounded annually. That investment is now worth $3,000. How old are you today?


a.
77

b.
84

c.
82

d.
86

e.
87

We can solve this problem using the formula for compound interest:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $ A $ is the amount of money accumulated after interest,
- $ P $ is the principal amount (the initial investment),
- $ r $ is the annual interest rate (decimal),
- $ n $ is the number of times the interest is compounded per year,
- $ t $ is the time the money is invested or borrowed for, in years.

Given:
- $ A = 3000 $,
- $ P = 100 $,
- $ r = 4.5\% = 0.045 $,
- $ n = 1 $ (since it's compounded annually).

We need to solve for $ t $, which will give us the number of years since your tenth birthday.

The formula simplifies to:

\[
3000 = 100 \left(1 + 0.045\right)^t
\]

This can be rewritten as:

\[
3000 = 100 (1.045)^t
\]

Dividing both sides by 100:

\[
30 = (1.045)^t
\]

Now, take the natural logarithm of both sides:

\[
\ln(30) = t \ln(1.045)
\]

Solving for $ t $:

\[
t = \frac{\ln(30)}{\ln(1.045)}
\]

Let's calculate this.The time $ t $ is approximately 77.27 years. Since this is the number of years since your tenth birthday, your current age is:

\[
\text{Age} = 10 + 77.27 \approx 87
\]

So, you are about 87 years old today.

The correct answer is **e. 87**.

Would you like more details or have any further questions?

Here are 5 related questions for practice:
1. How would the final amount change if the interest rate was 5%?
2. What happens to the value of $ t $ if interest is compounded semi-annually?
3. How long would it take to double your money at 4.5% interest, compounded annually?
4. If you invested $200 instead of $100, how old would you be today under the same conditions?
5. How does the compound interest formula differ from the simple interest formula?

**Tip:** When solving compound interest problems, always remember to adjust the interest rate and time period for different compounding intervals.

On your tenth birthday, you received $100 which you invested at 4.5 percent interest, compounded annually. That investment is now worth $3,000. How old are you today? a. 77, b. 84, c. 82, d. 86, e. 87

Find out how to calculate your age based on an initial $100 investment at 4.5% annual interest, compounded yearly, growing to $3,000. This problem covers compound interest, logarithmic calculations, and exponential growth, suitable for students in Grades 10-12.

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