Lisa McNeil needs to choose between two​ investments: One pays 11.6​% compounded continuously​, and the other pays 12​% compounded semiannually. If she plans to invest ​$10 comma 000 for 2 ​years, which investment should she​ choose? How much extra interest will she earn by making the better​ choice?

To determine which investment Lisa McNeil should choose, we need to compare the amount of money she will have after 2 years with each option. 

### 1. Continuous Compounding at 11.6%
For continuously compounded interest, the formula is:

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

Where:
- $ A $ is the future value of the investment
- $ P $ is the principal (initial investment) = \$10,000
- $ r $ is the interest rate = 11.6% = 0.116
- $ t $ is the time in years = 2 years
- $ e $ is the mathematical constant (approximately 2.71828)

We can calculate the future value using this formula.

### 2. Semiannual Compounding at 12%
For interest compounded semiannually, the formula is:

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

Where:
- $ A $ is the future value of the investment
- $ P $ is the principal (initial investment) = \$10,000
- $ r $ is the interest rate = 12% = 0.12
- $ n $ is the number of compounding periods per year = 2 (since it’s semiannual)
- $ t $ is the time in years = 2 years

Now, let's calculate both options.Here are the results:

1. **Continuous compounding at 11.6%**: After 2 years, Lisa will have approximately **\$12,611.20**.
2. **Semiannual compounding at 12%**: After 2 years, Lisa will have approximately **\$12,624.77**.

The **better choice** is the semiannual compounding at 12%, as it results in slightly more money.

By choosing the semiannual compounding option, Lisa will earn **\$13.57** more in interest over 2 years.

---

Would you like further details or have any questions? Here are some related questions:

1. What happens if the compounding frequency changes to monthly?
2. How do different investment time periods affect the outcome?
3. What are the benefits of continuous compounding over other methods?
4. How would a change in the initial investment affect the difference in interest earned?
5. How does compounding work when dealing with variable interest rates?

**Tip**: Always compare investments with the same compounding period for a more accurate comparison of interest rates.

Lisa McNeil needs to choose between two​ investments: One pays 11.6​% compounded continuously​, and the other pays 12​% compounded semiannually. If she plans to invest ​$10,000 for 2 ​years, which investment should she​ choose? How much extra interest will she earn by making the better​ choice?

Compare two investment options for Lisa McNeil: 11.6% compounded continuously versus 12% compounded semiannually for $10,000 over 2 years. Understand the key formulas for continuous and semiannual compounding, and see how much extra interest she can earn by choosing the better option.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation