Answer :
Given:
Initial deposit: $46,000
Annual interest rate: 3.8%
Compounding frequency: monthly
Monthly withdrawal: $1,000
Time period: 1 year (12 months)
Step 1: Calculate the monthly interest rate
The annual interest rate is 3.8%, which compounds monthly. Therefore, the monthly interest rate
r is:
=
3.8
%
12
=
0.038
12
=
0.0031667
r=
12
3.8%
β
=
12
0.038
β
=0.0031667
Step 2: Calculate the balance after each month
Janine starts with $46,000 and withdraws $1,000 at the end of each month. We need to calculate the balance at the end of each month considering the interest and withdrawals.
Let's denote the balance at the end of the
n-th month as
A
n
β
.
Initial amount:
0
=
46000
A
0
β
=46000
For each month
n (from 1 to 12):
=
(
β
1
Γ
(
1
+
)
)
β
1000
A
n
β
=(A
nβ1
β
Γ(1+r))β1000
We'll calculate this iteratively for each month.
Month 1:
1
=
(
46000
Γ
(
1
+
0.0031667
)
)
β
1000
A
1
β
=(46000Γ(1+0.0031667))β1000
1
=
(
46000
Γ
1.0031667
)
β
1000
A
1
β
=(46000Γ1.0031667)β1000
1
=
46145.6662
β
1000
A
1
β
=46145.6662β1000
1
β
45145.67
A
1
β
β45145.67
Month 2:
2
=
(
45145.67
Γ
1.0031667
)
β
1000
A
2
β
=(45145.67Γ1.0031667)β1000
2
=
45288.1275
β
1000
A
2
β
=45288.1275β1000
2
β
44288.13
A
2
β
β44288.13
Month 3:
3
=
(
44288.13
Γ
1.0031667
)
β
1000
A
3
β
=(44288.13Γ1.0031667)β1000
3
=
44427.4715
β
1000
A
3
β
=44427.4715β1000
3
β
43427.47
A
3
β
β43427.47
Month 4:
4
=
(
43427.47
Γ
1.0031667
)
β
1000
A
4
β
=(43427.47Γ1.0031667)β1000
4
=
43563.6902
β
1000
A
4
β
=43563.6902β1000
4
β
42563.69
A
4
β
β42563.69
Month 5:
5
=
(
42563.69
Γ
1.0031667
)
β
1000
A
5
β
=(42563.69Γ1.0031667)β1000
5
=
42696.7823
β
1000
A
5
β
=42696.7823β1000
5
β
41696.78
A
5
β
β41696.78
Month 6:
6
=
(
41696.78
Γ
1.0031667
)
β
1000
A
6
β
=(41696.78Γ1.0031667)β1000
6
=
41826.7444
β
1000
A
6
β
=41826.7444β1000
6
β
40826.74
A
6
β
β40826.74
Month 7:
7
=
(
40826.74
Γ
1.0031667
)
β
1000
A
7
β
=(40826.74Γ1.0031667)β1000
7
=
40953.5722
β
1000
A
7
β
=40953.5722β1000
7
β
39953.57
A
7
β
β39953.57
Month 8:
8
=
(
39953.57
Γ
1.0031667
)
β
1000
A
8
β
=(39953.57Γ1.0031667)β1000
8
=
40077.2624
β
1000
A
8
β
=40077.2624β1000
8
β
39077.26
A
8
β
β39077.26
Month 9:
9
=
(
39077.26
Γ
1.0031667
)
β
1000
A
9
β
=(39077.26Γ1.0031667)β1000
9
=
39197.8117
β
1000
A
9
β
=39197.8117β1000
9
β
38197.81
A
9
β
β38197.81
Month 10:
10
=
(
38197.81
Γ
1.0031667
)
β
1000
A
10
β
=(38197.81Γ1.0031667)β1000
10
=
38315.2176
β
1000
A
10
β
=38315.2176β1000
10
β
37315.22
A
10
β
β37315.22
Month 11:
11
=
(
37315.22
Γ
1.0031667
)
β
1000
A
11
β
=(37315.22Γ1.0031667)β1000
11
=
37429.4764
β
1000
A
11
β
=37429.4764β1000
11
β
36429.48
A
11
β
β36429.48
Month 12:
12
=
(
36429.48
Γ
1.0031667
)
β
1000
A
12
β
=(36429.48Γ1.0031667)β1000
12
=
36540.5857
β
1000
A
12
β
=36540.5857β1000
12
β
35540.59
A
12
β
β35540.59
Step 3: Calculate the total interest earned
The total interest earned is the difference between the total amount in the account after one year (with monthly withdrawals) and the total amount deposited minus the total withdrawals.
Total amount deposited initially:
46000
46000
Total withdrawals over the year:
1000
Γ
12
=
12000
1000Γ12=12000
Total amount at the end of the year:
12
β
35540.59
A
12
β
β35540.59
Total interest earned:
Interest
=
12
β
(
46000
β
12000
)
Interest=A
12
β
β(46000β12000)
Interest
=
35540.59
β
34000
Interest=35540.59β34000
Interest
=
1554.59
Interest=1554.59
The closest answer to $1554.59 is:
Answer:
A $1568