https://file.aiquickdraw.com/mathbot/file/17292627266749ja3spyd.jpg

The document you've uploaded is a worksheet comparing simple and compound interest, with the following investment scenario:

- You have $3000 to invest as a lump sum.
- Two options are given:
  1. **3% simple interest**
  2. **2.5% compound interest**
  
The worksheet asks you to calculate how much the investment will be worth after **10 years** and **20 years**, using both simple interest and compound interest formulas.

Let's go through both options for each period.

### Formulas:
1. **Simple Interest Formula**:  
   \[
   A = P(1 + rt)
   \]
   where:
   - $ P $ = Principal (initial investment),
   - $ r $ = Interest rate (as a decimal),
   - $ t $ = Time (in years),
   - $ A $ = Amount after interest.

2. **Compound Interest Formula**:  
   \[
   A = P \left(1 + \frac{r}{n}\right)^{nt}
   \]
   where:
   - $ P $ = Principal,
   - $ r $ = Interest rate (as a decimal),
   - $ t $ = Time (in years),
   - $ n $ = Number of times interest is compounded per year (assume annually unless stated otherwise),
   - $ A $ = Amount after interest.

### 1. For 10 Years:

#### Option A: Simple Interest (3%)
- $ P = 3000 $
- $ r = 0.03 $ (3% interest)
- $ t = 10 $ years

\[
A = 3000(1 + 0.03 \times 10) = 3000(1 + 0.3) = 3000 \times 1.3 = 3900
\]
The investment will be worth **$3900** after 10 years with simple interest.

#### Option B: Compound Interest (2.5%)
- $ P = 3000 $
- $ r = 0.025 $ (2.5% interest)
- $ t = 10 $ years
- Assume $ n = 1 $ (compounded annually)

\[
A = 3000 \left(1 + \frac{0.025}{1}\right)^{1 \times 10} = 3000(1.025)^{10}
\]
\[
A \approx 3000 \times 1.28008 = 3840.24
\]
The investment will be worth **$3840.24** after 10 years with compound interest.

### 2. For 20 Years:

#### Option A: Simple Interest (3%)
- $ P = 3000 $
- $ r = 0.03 $
- $ t = 20 $ years

\[
A = 3000(1 + 0.03 \times 20) = 3000(1 + 0.6) = 3000 \times 1.6 = 4800
\]
The investment will be worth **$4800** after 20 years with simple interest.

#### Option B: Compound Interest (2.5%)
- $ P = 3000 $
- $ r = 0.025 $
- $ t = 20 $ years
- Assume $ n = 1 $ (compounded annually)

\[
A = 3000 \left(1 + \frac{0.025}{1}\right)^{1 \times 20} = 3000(1.025)^{20}
\]
\[
A \approx 3000 \times 1.63862 = 4915.87
\]
The investment will be worth **$4915.87** after 20 years with compound interest.

### Summary of Results:

| Years | Simple Interest (3%) | Compound Interest (2.5%) |
|-------|----------------------|--------------------------|
| 10    | $3900                | $3840.24                 |
| 20    | $4800                | $4915.87                 |

### Which is better?
- After **10 years**, simple interest gives a slightly higher return ($3900 vs. $3840.24).
- After **20 years**, compound interest gives a slightly higher return ($4915.87 vs. $4800).

In general, compound interest tends to outperform simple interest over longer periods, as it grows exponentially, while simple interest grows linearly.

Would you like further explanation or calculations?

---

**Follow-up Questions:**
1. How does compound interest outperform simple interest in the long run?
2. What if the interest was compounded quarterly or monthly?
3. How does the interest rate affect the comparison between simple and compound interest?
4. Could reinvesting the interest periodically impact the returns in real-life scenarios?
5. What other factors might influence the decision between simple and compound interest investments?

**Tip**: Compound interest becomes more advantageous as the investment period or compounding frequency increases.

You have $3000 to invest as a lump sum, with two investment options: 3% simple interest or 2.5% compound interest. The worksheet asks to determine how much the investment will be worth after 10 years and 20 years for both options.

Explore the comparison between 3% simple interest and 2.5% compound interest on a $3000 investment. This problem demonstrates how interest grows over 10 and 20 years, using key financial formulas. Suitable for high school students learning about interest calculation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation