https://file.aiquickdraw.com/mathbot/file/1730213785956yj4ryokj.jpg

Let's tackle each question one at a time.

---

### Question 1:
We are given:
- Principal (P) = $4000
- Annual simple interest rate (r) = 2%
- Time (t) = 10 years

For simple interest, the formula is:
\[
\text{Interest} = P \times r \times t
\]

1. Convert the interest rate to decimal form: $ r = \frac{2}{100} = 0.02 $.
2. Plug the values into the formula:
   \[
   \text{Interest} = 4000 \times 0.02 \times 10 = 4000 \times 0.2 = 800
   \]
   
So, **the interest earned after 10 years is $800**.

---

### Question 2:
We are given:
- Principal (P) = $2000
- Annual interest rate (APR) = 6%
- Time (t) = 20 years

This question involves two parts:
1. **Simple Interest Calculation**:
   \[
   \text{Amount (A)} = P + (\text{Interest})
   \]
   where:
   \[
   \text{Interest} = P \times r \times t
   \]

   Converting the rate to decimal:
   \[
   r = 0.06
   \]
   Then:
   \[
   \text{Interest} = 2000 \times 0.06 \times 20 = 2000 \times 1.2 = 2400
   \]
   Therefore, the **balance with simple interest** would be:
   \[
   A = 2000 + 2400 = 4400
   \]

2. **Compound Interest Calculation (Compounded Monthly)**:
   For compound interest, the formula is:
   \[
   A = P \times \left(1 + \frac{r}{n}\right)^{n \times t}
   \]
   where:
   - $ P = 2000 $
   - $ r = 0.06 $
   - $ t = 20 $ years
   - $ n = 12 $ (monthly compounding)

   Plugging in the values:
   \[
   A = 2000 \times \left(1 + \frac{0.06}{12}\right)^{12 \times 20}
   \]
   Calculating the monthly rate:
   \[
   \frac{0.06}{12} = 0.005
   \]
   Then:
   \[
   A = 2000 \times (1 + 0.005)^{240}
   \]
   Now we calculate $ (1.005)^{240} $ and then multiply by 2000.

Using a calculator, $ (1.005)^{240} \approx 3.3102 $. Therefore:
\[
A \approx 2000 \times 3.3102 = 6620.40
\]

So, **the balance with monthly compounding after 20 years is approximately $6620.40**.

---

Would you like a detailed breakdown of each formula or further explanations on interest calculations?

### Further Questions to Explore
1. How would the answer change if the interest were compounded quarterly?
2. What if the APR were different, say 5% or 8%? How would it affect both simple and compound interest outcomes?
3. What if we invested a different principal amount, say $3000, under the same conditions?
4. How would the calculations change if interest were compounded daily?
5. Could we use these calculations to compare different banks' offerings?

### Tip
When using compound interest formulas, ensure you adjust the rate and time period according to the compounding frequency (monthly, quarterly, etc.) to get an accurate result.

1. We invest $4000 in an account that pays simple interest of 2% each year. How much interest is earned after 10 years?

2. Suppose you invest $2000 in a savings account that pays an APR of 6%. If the account pays simple interest, what is the balance in the account after 20 years? (Round your answer to the nearest cent.)

If interest is compounded monthly, what is the balance in the account after 20 years? (Round your answer to the nearest cent.)

Solve problems involving simple and compound interest. Learn how to calculate interest on a $4000 investment at 2% simple interest over 10 years and a $2000 investment with 6% APR compounded monthly over 20 years. Perfect for high school students studying interest calculations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation