Question
I have $900 to invest in a savings account with a 6% annual interest rate compounded monthly. the bank uses 360 days in a year for calculation. Can you find the formula to calculate the total amount after n days and compute the total amount after 30, 60 and 90 days?
Answers
GPT-4o mini
To calculate the total amount in a savings account with compound interest, you can use the formula:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \(A\) is the amount of money accumulated after time \(t\),
- \(P\) is the principal amount (the initial amount of money),
- \(r\) is the annual interest rate (decimal),
- \(n\) is the number of times that interest is compounded per year,
- \(t\) is the time in years.
In your case:
- \(P = 900\),
- \(r = 0.06\) (which is 6% as a decimal),
- \(n = 12\) (since interest is compounded monthly).
Since the bank uses a 360-day year for interest calculation, when calculating the time \(t\) in years for a given number of days, use:
\[
t = \frac{\text{number of days}}{360}
\]
Now, let's compute the total amount after 30, 60, and 90 days.
### Step 1: Calculate for 30 days
1. Calculate \(t\):
\[
t = \frac{30}{360} = \frac{1}{12} \text{ years}
\]
2. Substitute into the formula:
\[
A = 900 \left(1 + \frac{0.06}{12}\right)^{12 \cdot \frac{1}{12}}
\]
3. Calculate:
\[
A = 900 \left(1 + 0.005\right)^{1}
\]
\[
A = 900 \cdot 1.005 = 904.50
\]
### Step 2: Calculate for 60 days
1. Calculate \(t\):
\[
t = \frac{60}{360} = \frac{1}{6} \text{ years}
\]
2. Substitute into the formula:
\[
A = 900 \left(1 + \frac{0.06}{12}\right)^{12 \cdot \frac{1}{6}}
\]
3. Calculate:
\[
A = 900 \left(1 + 0.005\right)^{2}
\]
\[
A = 900 \cdot (1.005)^{2} = 900 \cdot 1.010025 = 909.02
\]
### Step 3: Calculate for 90 days
1. Calculate \(t\):
\[
t = \frac{90}{360} = \frac{1}{4} \text{ years}
\]
2. Substitute into the formula:
\[
A = 900 \left(1 + \frac{0.06}{12}\right)^{12 \cdot \frac{1}{4}}
\]
3. Calculate:
\[
A = 900 \left(1 + 0.005\right)^{3}
\]
\[
A = 900 \cdot (1.005)^{3} = 900 \cdot 1.015075 = 913.57
\]
### Summary of Results:
- After 30 days: **$904.50**
- After 60 days: **$909.02**
- After 90 days: **$913.57**
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \(A\) is the amount of money accumulated after time \(t\),
- \(P\) is the principal amount (the initial amount of money),
- \(r\) is the annual interest rate (decimal),
- \(n\) is the number of times that interest is compounded per year,
- \(t\) is the time in years.
In your case:
- \(P = 900\),
- \(r = 0.06\) (which is 6% as a decimal),
- \(n = 12\) (since interest is compounded monthly).
Since the bank uses a 360-day year for interest calculation, when calculating the time \(t\) in years for a given number of days, use:
\[
t = \frac{\text{number of days}}{360}
\]
Now, let's compute the total amount after 30, 60, and 90 days.
### Step 1: Calculate for 30 days
1. Calculate \(t\):
\[
t = \frac{30}{360} = \frac{1}{12} \text{ years}
\]
2. Substitute into the formula:
\[
A = 900 \left(1 + \frac{0.06}{12}\right)^{12 \cdot \frac{1}{12}}
\]
3. Calculate:
\[
A = 900 \left(1 + 0.005\right)^{1}
\]
\[
A = 900 \cdot 1.005 = 904.50
\]
### Step 2: Calculate for 60 days
1. Calculate \(t\):
\[
t = \frac{60}{360} = \frac{1}{6} \text{ years}
\]
2. Substitute into the formula:
\[
A = 900 \left(1 + \frac{0.06}{12}\right)^{12 \cdot \frac{1}{6}}
\]
3. Calculate:
\[
A = 900 \left(1 + 0.005\right)^{2}
\]
\[
A = 900 \cdot (1.005)^{2} = 900 \cdot 1.010025 = 909.02
\]
### Step 3: Calculate for 90 days
1. Calculate \(t\):
\[
t = \frac{90}{360} = \frac{1}{4} \text{ years}
\]
2. Substitute into the formula:
\[
A = 900 \left(1 + \frac{0.06}{12}\right)^{12 \cdot \frac{1}{4}}
\]
3. Calculate:
\[
A = 900 \left(1 + 0.005\right)^{3}
\]
\[
A = 900 \cdot (1.005)^{3} = 900 \cdot 1.015075 = 913.57
\]
### Summary of Results:
- After 30 days: **$904.50**
- After 60 days: **$909.02**
- After 90 days: **$913.57**