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.
•015
T=p.r.t
• Account I earns 1,5% annual simple interest.
° Account II earns 1.5% interest compounded annually. A = P(I + r)E
Which amount is closest to the difference between the balances of the two accounts at the end of 4 years?
A $1.64
© $72.00
© $73.64
© $1.80
•015
T=p.r.t
• Account I earns 1,5% annual simple interest.
° Account II earns 1.5% interest compounded annually. A = P(I + r)E
Which amount is closest to the difference between the balances of the two accounts at the end of 4 years?
A $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 need to calculate the final amounts in each account using the respective interest formulas.
### Account I: Simple Interest
The formula for simple interest is:
\[ T = P \cdot r \cdot t \]
Where:
- \( T \) is the total interest earned,
- \( P \) is the principal (initial deposit),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time in years.
Given:
- \( P = 1200 \)
- \( r = 0.015 \) (1.5% as a decimal)
- \( t = 4 \)
Calculating the interest:
\[ T = 1200 \cdot 0.015 \cdot 4 \]
\[ T = 1200 \cdot 0.06 \]
\[ T = 72 \]
So, the total amount in Account I after 4 years:
\[ A_I = P + T = 1200 + 72 = 1272 \]
### Account II: Compound Interest
The formula for compound interest is:
\[ A = P \cdot (1 + r)^t \]
Where:
- \( A \) is the total amount after time \( t \),
- \( P \) is the principal,
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time in years.
Given:
- \( P = 1200 \)
- \( r = 0.015 \)
- \( t = 4 \)
Calculating the amount:
\[ A_{II} = 1200 \cdot (1 + 0.015)^4 \]
\[ A_{II} = 1200 \cdot (1.015)^4 \]
Calculating \( (1.015)^4 \):
\[ (1.015)^4 \approx 1.061364 \]
Now, substituting this value back:
\[ A_{II} \approx 1200 \cdot 1.061364 \]
\[ A_{II} \approx 1273.64 \]
### Difference Between the Two Accounts
Now, we find the difference:
\[ \text{Difference} = A_{II} - A_I \]
\[ \text{Difference} = 1273.64 - 1272 \]
\[ \text{Difference} \approx 1.64 \]
Thus, the amount closest to the difference between the balances of the two accounts at the end of 4 years is **$1.64**.
The answer is:
**$1.64**
### Account I: Simple Interest
The formula for simple interest is:
\[ T = P \cdot r \cdot t \]
Where:
- \( T \) is the total interest earned,
- \( P \) is the principal (initial deposit),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time in years.
Given:
- \( P = 1200 \)
- \( r = 0.015 \) (1.5% as a decimal)
- \( t = 4 \)
Calculating the interest:
\[ T = 1200 \cdot 0.015 \cdot 4 \]
\[ T = 1200 \cdot 0.06 \]
\[ T = 72 \]
So, the total amount in Account I after 4 years:
\[ A_I = P + T = 1200 + 72 = 1272 \]
### Account II: Compound Interest
The formula for compound interest is:
\[ A = P \cdot (1 + r)^t \]
Where:
- \( A \) is the total amount after time \( t \),
- \( P \) is the principal,
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time in years.
Given:
- \( P = 1200 \)
- \( r = 0.015 \)
- \( t = 4 \)
Calculating the amount:
\[ A_{II} = 1200 \cdot (1 + 0.015)^4 \]
\[ A_{II} = 1200 \cdot (1.015)^4 \]
Calculating \( (1.015)^4 \):
\[ (1.015)^4 \approx 1.061364 \]
Now, substituting this value back:
\[ A_{II} \approx 1200 \cdot 1.061364 \]
\[ A_{II} \approx 1273.64 \]
### Difference Between the Two Accounts
Now, we find the difference:
\[ \text{Difference} = A_{II} - A_I \]
\[ \text{Difference} = 1273.64 - 1272 \]
\[ \text{Difference} \approx 1.64 \]
Thus, the amount closest to the difference between the balances of the two accounts at the end of 4 years is **$1.64**.
The answer is:
**$1.64**