Question

Understanding Interest Practice
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4 of 54 of 5 Items
Question
Determine whether an 8.3% simple interest rate or a 7.2% compound interest rate with interest compounded monthly is the better investing option when $3,800 is invested for 4 years.

Enter 1 if an 8.3% simple interest rate is the better option.

Enter 2 if a 7.2% compound interest rate with interest compounded monthly is the better option.

(1 point)
Remaining Attempts : 3

Answer 1

To determine which is the better investment option, we will calculate the future value for both the simple interest and the compound interest scenarios.

### Option 1: Simple Interest
The formula for simple interest is:
\[ A = P(1 + rt) \]
Where:
- \( A \) = the total amount of money after interest
- \( P \) = principal amount (initial investment)
- \( r \) = annual interest rate (as a decimal)
- \( t \) = time in years

For the simple interest:
- \( P = 3,800 \)
- \( r = 8.3\% = 0.083 \)
- \( t = 4 \)

Calculating:
\[ A = 3,800(1 + (0.083)(4)) \]
\[ A = 3,800(1 + 0.332) \]
\[ A = 3,800(1.332) \]
\[ A \approx 5,063.60 \]

### Option 2: Compound Interest
The formula for compound interest is:
\[ A = P\left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) = the total amount of money after interest
- \( P \) = principal amount (initial investment)
- \( r \) = annual interest rate (as a decimal)
- \( n \) = number of times interest is compounded per year
- \( t \) = time in years

For the compound interest:
- \( P = 3,800 \)
- \( r = 7.2\% = 0.072 \)
- \( n = 12 \) (compounded monthly)
- \( t = 4 \)

Calculating:
\[ A = 3,800\left(1 + \frac{0.072}{12}\right)^{12 \cdot 4} \]
\[ A = 3,800\left(1 + 0.006\right)^{48} \]
\[ A = 3,800\left(1.006\right)^{48} \]
\[ A \approx 3,800 \times 1.34885 \]
\[ A \approx 5,120.43 \]

### Conclusion
- Future value with simple interest: **$5,063.60**
- Future value with compound interest: **$5,120.43**

Since **$5,120.43** (compound interest) is greater than **$5,063.60** (simple interest), the better option is:
**Enter 2** (7.2% compound interest rate compounded monthly).