A deposit of $3,800 is made into each of two bank accounts. No additional deposits or withdrawals will be made for 5 years.

Account I earns 4.2% annual simple interest.
Account II earns 4.2% interest compounded annually.

Which amount is closest to the difference between the balances of the two accounts at the end of 5 years?
A.$69.91
B.$106.91
C.$165.91
D.$6.91

We can start by calculating the balances of both accounts at the end of 5 years, using the formulas for **simple interest** and **compound interest**, and then determine the difference between the two balances.

### **Account I: Simple Interest**
The formula for simple interest is:

\[
A = P(1 + rt)
\]
where:
- $A$ is the final amount (balance),
- $P = 3800$ is the principal (initial deposit),
- $r = 0.042$ is the annual interest rate (4.2%),
- $t = 5$ is the time in years.

Substitute the values:

\[
A = 3800(1 + 0.042 \times 5) = 3800(1 + 0.21) = 3800 \times 1.21 = 4598
\]

So, the balance in **Account I** at the end of 5 years is **$4,598**.

### **Account II: Compound Interest**
The formula for compound interest is:

\[
A = P\left(1 + \frac{r}{n}\right)^{nt}
\]
where:
- $A$ is the final amount (balance),
- $P = 3800$,
- $r = 0.042$,
- $n = 1$ (since interest is compounded annually),
- $t = 5$.

Substitute the values:

\[
A = 3800\left(1 + \frac{0.042}{1}\right)^{1 \times 5} = 3800(1.042)^5
\]

Now, calculate $1.042^5$:

\[
1.042^5 \approx 1.229
\]

So,

\[
A = 3800 \times 1.229 = 4670.2
\]

The balance in **Account II** at the end of 5 years is approximately **$4,670.20**.

### **Difference Between Balances**
Now, calculate the difference between the balances of Account II and Account I:

\[
4670.20 - 4598 = 72.20
\]

The closest amount to the difference is **$69.91**.

### **Answer: A. $69.91**

Would you like further details or have any questions?

Here are five related questions you might explore:
1. How would the difference change if the interest rate was higher, say 5%?
2. What would the balances be after 10 years with the same interest rates?
3. How does the frequency of compounding affect the final amount?
4. What would happen if withdrawals were made annually?
5. How would the result differ if the deposit was doubled?

**Tip:** Compound interest grows faster than simple interest over time because you earn interest on both the initial deposit and the previously accumulated interest.

Calculate the difference between simple and compound interest for a $3,800 deposit over 5 years. Account I earns 4.2% simple interest, while Account II earns 4.2% compounded annually. Learn which answer is closest to the difference, $69.91, using detailed formulas.

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