Question
A deposit of $1,200 is made into each of two bank accounts. No additional deposits or withdrawals will be made for 4 years.
• Account I earns 1.5% annual simple interest.
• Account II earns 1.5% interest compounded annually.
Which amount is closest to the difference between the balances of the two accounts at the end of 4 years?
$1.64
$72.00
$73.64
$1.80
• Account I earns 1.5% annual simple interest.
• Account II earns 1.5% interest compounded annually.
Which amount is closest to the difference between the balances of the two accounts at the end of 4 years?
$1.64
$72.00
$73.64
$1.80
Answers
GPT-4o mini
To find the difference between the balances of the two accounts after 4 years, we first need to calculate the final balance for each account.
### Account I (Simple Interest)
Using the formula for simple interest:
\[ A = P(1 + rt) \]
where:
- \( A \) is the total amount after time \( t \),
- \( P \) is the principal amount (initial deposit),
- \( r \) is the rate of interest per year (as a decimal),
- \( t \) is the time the money is invested for in years.
For Account I:
- \( P = 1200 \)
- \( r = 0.015 \) (1.5%)
- \( t = 4 \)
Calculating the total amount:
\[ A = 1200(1 + 0.015 \cdot 4) \]
\[ A = 1200(1 + 0.06) \]
\[ A = 1200(1.06) \]
\[ A = 1272 \]
So, the balance in Account I after 4 years is **$1,272**.
### Account II (Compound Interest)
Using the formula for compound interest:
\[ A = P(1 + r)^t \]
For Account II:
- \( P = 1200 \)
- \( r = 0.015 \)
- \( t = 4 \)
Calculating the total amount:
\[ A = 1200(1 + 0.015)^4 \]
\[ A = 1200(1.015)^4 \]
Now we can calculate \( (1.015)^4 \):
\[ (1.015)^4 \approx 1.061364 \] (using a calculator)
Now substituting back into the formula:
\[ A \approx 1200 \times 1.061364 \]
\[ A \approx 1273.64 \]
So, the balance in Account II after 4 years is approximately **$1,273.64**.
### Difference Between the Balances
Now, we can find the difference between the two accounts:
\[ \text{Difference} = A_{\text{II}} - A_{\text{I}} \]
\[ = 1273.64 - 1272 \]
\[ = 1.64 \]
Thus, the amount closest to the difference between the balances of the two accounts at the end of 4 years is **$1.64**.
### Account I (Simple Interest)
Using the formula for simple interest:
\[ A = P(1 + rt) \]
where:
- \( A \) is the total amount after time \( t \),
- \( P \) is the principal amount (initial deposit),
- \( r \) is the rate of interest per year (as a decimal),
- \( t \) is the time the money is invested for in years.
For Account I:
- \( P = 1200 \)
- \( r = 0.015 \) (1.5%)
- \( t = 4 \)
Calculating the total amount:
\[ A = 1200(1 + 0.015 \cdot 4) \]
\[ A = 1200(1 + 0.06) \]
\[ A = 1200(1.06) \]
\[ A = 1272 \]
So, the balance in Account I after 4 years is **$1,272**.
### Account II (Compound Interest)
Using the formula for compound interest:
\[ A = P(1 + r)^t \]
For Account II:
- \( P = 1200 \)
- \( r = 0.015 \)
- \( t = 4 \)
Calculating the total amount:
\[ A = 1200(1 + 0.015)^4 \]
\[ A = 1200(1.015)^4 \]
Now we can calculate \( (1.015)^4 \):
\[ (1.015)^4 \approx 1.061364 \] (using a calculator)
Now substituting back into the formula:
\[ A \approx 1200 \times 1.061364 \]
\[ A \approx 1273.64 \]
So, the balance in Account II after 4 years is approximately **$1,273.64**.
### Difference Between the Balances
Now, we can find the difference between the two accounts:
\[ \text{Difference} = A_{\text{II}} - A_{\text{I}} \]
\[ = 1273.64 - 1272 \]
\[ = 1.64 \]
Thus, the amount closest to the difference between the balances of the two accounts at the end of 4 years is **$1.64**.