Marie wants to have P20,000 in two years.
The following rates are available:

a. 6.2% simple interest
b. 6.1% compounded annually
c. 5.8% compounded semiannually

Which account should she choose if she warts to
invest the smallest amount of money now?

How much money must he invest to accumulate
P20,000 in two years' time?

To determine which account Marie should choose to invest the smallest amount of money now, we need to calculate the **present value** (the amount she needs to invest today) for each option.

### 1. **Simple Interest Formula:**
The formula for simple interest is:

\[
A = P(1 + rt)
\]

Where:
- \( A \) is the amount of money accumulated after \( t \) years (including interest),
- \( P \) is the principal amount (initial investment),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time the money is invested for (in years).

Rearranging the formula to solve for \( P \):

\[
P = \frac{A}{1 + rt}
\]

For option **a** (6.2% simple interest):

\[
P = \frac{20,000}{1 + (0.062 \times 2)} = \frac{20,000}{1.124} \approx 17,803.57
\]

### 2. **Compound Interest Formula (Compounded Annually):**
The formula for compound interest is:

\[
A = P(1 + r)^t
\]

Rearranging to solve for \( P \):

\[
P = \frac{A}{(1 + r)^t}
\]

For option **b** (6.1% compounded annually):

\[
P = \frac{20,000}{(1 + 0.061)^2} = \frac{20,000}{1.1257} \approx 17,765.54
\]

### 3. **Compound Interest Formula (Compounded Semiannually):**
For compounding periods other than annually, the formula is:

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

Where:
- \( n \) is the number of times the interest is compounded per year.

For option **c** (5.8% compounded semiannually, \( n = 2 \)):

\[
P = \frac{20,000}{\left(1 + \frac{0.058}{2}\right)^{2 \times 2}} = \frac{20,000}{(1.029)^4} = \frac{20,000}{1.1216} \approx 17,828.37
\]

### Summary of Required Investments:
- **Option a (6.2% simple interest):** P17,803.57
- **Option b (6.1% compounded annually):** P17,765.54
- **Option c (5.8% compounded semiannually):** P17,828.37

### Conclusion:
Marie should choose **option b (6.1% compounded annually)** because it requires the smallest investment today, which is **P17,765.54**.

---

Would you like further details or explanations?

Here are some related questions:
1. How does simple interest differ from compound interest?
2. What happens if the compounding frequency increases?
3. How can we calculate the future value of an investment over more than two years?
4. What is continuous compounding, and how does it compare with these options?
5. How can Marie evaluate the effect of inflation on her investment returns?

**Tip:** The more frequently interest is compounded, the more interest you earn over time, but at the cost of potentially higher initial investments.

Marie wants to have P20,000 in two years. The following rates are available: a. 6.2% simple interest b. 6.1% compounded annually c. 5.8% compounded semiannually Which account should she choose if she wants to invest the smallest amount of money now? How much money must she invest to accumulate P20,000 in two years' time?

Marie needs P20,000 in two years and has to choose between 6.2% simple interest, 6.1% compounded annually, and 5.8% compounded semiannually. Learn which option requires the smallest initial investment and how to calculate the present value using simple and compound interest formulas.

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