Question
Round any interest to the nearest penny. Your fixed expenses are $1763.25/month. You saved 6 months worth of an emergency fund in a savings account earning 4.5% over 3 years. After 3 years, you withdrew $4360 ,,because of losing your job. What is your balance after the withdrawal? How much interest did you lose because of the withdrawal?
Answers
GPT-4o mini
To find your balance after the withdrawal and how much interest you lost because of the withdrawal, we can break the problem down into steps.
### Step 1: Calculate the emergency fund amount
You saved 6 months' worth of fixed expenses:
\[
\text{Emergency Fund} = 6 \times 1763.25 = 10579.50
\]
### Step 2: Calculate the interest earned over 3 years
To calculate the interest earned on your savings account, we can use the formula for compound interest:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \(A\) = the amount of money accumulated after n years, including interest.
- \(P\) = principal amount (initial investment): $10,579.50
- \(r\) = annual interest rate (decimal): 0.045
- \(n\) = number of times that interest is compounded per year: 1 (assuming it's compounded annually)
- \(t\) = number of years the money is invested: 3
Plugging in the values:
\[
A = 10579.50 \left(1 + \frac{0.045}{1}\right)^{1 \times 3}
\]
\[
A = 10579.50 \left(1.045\right)^{3}
\]
Calculating \(1.045^3\):
\[
1.045^3 \approx 1.1407
\]
Now plug this back into the equation:
\[
A \approx 10579.50 \times 1.1407 \approx 12073.85
\]
### Step 3: Calculate the balance after withdrawal
After withdrawing $4,360:
\[
\text{Balance after withdrawal} = A - \text{Withdrawal}
\]
\[
\text{Balance after withdrawal} = 12073.85 - 4360 = 7703.85
\]
### Step 4: Calculate how much interest was lost because of the withdrawal
First, let’s calculate how much interest the original amount ($12,073.85) would have earned if it were not withdrawn at the end of the 3 years. If we assume the same interest for another year (compounding):
\[
\text{Interest if not withdrawn} = 12073.85 \times 0.045 \approx 543.32
\]
\[
\text{New total if not withdrawn} = 12073.85 + 543.32 \approx 12617.17
\]
Now, calculate how much interest you lose because of the withdrawal:
1. Compute the interest on the remaining balance after the withdrawal for one year:
\[
\text{Interest on remaining balance for next year} = 7703.85 \times 0.045 \approx 346.66
\]
The total interest you would have had without withdrawal:
\[
12617.17 - 7703.85 = 4913.32
\]
2. The lost interest due to withdrawal:
\[
\text{Lost Interest} = 543.32 - 346.66 \approx 196.66
\]
### Final Results
- **Balance after the withdrawal:** **$7,703.85**
- **Interest lost due to the withdrawal:** **$196.66**
After rounding to the nearest penny:
- Balance: **$7,703.85**
- Interest lost: **$196.66**
### Step 1: Calculate the emergency fund amount
You saved 6 months' worth of fixed expenses:
\[
\text{Emergency Fund} = 6 \times 1763.25 = 10579.50
\]
### Step 2: Calculate the interest earned over 3 years
To calculate the interest earned on your savings account, we can use the formula for compound interest:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \(A\) = the amount of money accumulated after n years, including interest.
- \(P\) = principal amount (initial investment): $10,579.50
- \(r\) = annual interest rate (decimal): 0.045
- \(n\) = number of times that interest is compounded per year: 1 (assuming it's compounded annually)
- \(t\) = number of years the money is invested: 3
Plugging in the values:
\[
A = 10579.50 \left(1 + \frac{0.045}{1}\right)^{1 \times 3}
\]
\[
A = 10579.50 \left(1.045\right)^{3}
\]
Calculating \(1.045^3\):
\[
1.045^3 \approx 1.1407
\]
Now plug this back into the equation:
\[
A \approx 10579.50 \times 1.1407 \approx 12073.85
\]
### Step 3: Calculate the balance after withdrawal
After withdrawing $4,360:
\[
\text{Balance after withdrawal} = A - \text{Withdrawal}
\]
\[
\text{Balance after withdrawal} = 12073.85 - 4360 = 7703.85
\]
### Step 4: Calculate how much interest was lost because of the withdrawal
First, let’s calculate how much interest the original amount ($12,073.85) would have earned if it were not withdrawn at the end of the 3 years. If we assume the same interest for another year (compounding):
\[
\text{Interest if not withdrawn} = 12073.85 \times 0.045 \approx 543.32
\]
\[
\text{New total if not withdrawn} = 12073.85 + 543.32 \approx 12617.17
\]
Now, calculate how much interest you lose because of the withdrawal:
1. Compute the interest on the remaining balance after the withdrawal for one year:
\[
\text{Interest on remaining balance for next year} = 7703.85 \times 0.045 \approx 346.66
\]
The total interest you would have had without withdrawal:
\[
12617.17 - 7703.85 = 4913.32
\]
2. The lost interest due to withdrawal:
\[
\text{Lost Interest} = 543.32 - 346.66 \approx 196.66
\]
### Final Results
- **Balance after the withdrawal:** **$7,703.85**
- **Interest lost due to the withdrawal:** **$196.66**
After rounding to the nearest penny:
- Balance: **$7,703.85**
- Interest lost: **$196.66**