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Sagot :
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%
ā
=
12
0.038
ā
=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
ā
.
Initial amount:
0
=
46000
A
0
ā
=46000
For each month
n (from 1 to 12):
=
(
ā
1
Ć
(
1
+
)
)
ā
1000
A
n
ā
=(A
nā1
ā
Ć(1+r))ā1000
We'll calculate this iteratively for each month.
Month 1:
1
=
(
46000
Ć
(
1
+
0.0031667
)
)
ā
1000
A
1
ā
=(46000Ć(1+0.0031667))ā1000
1
=
(
46000
Ć
1.0031667
)
ā
1000
A
1
ā
=(46000Ć1.0031667)ā1000
1
=
46145.6662
ā
1000
A
1
ā
=46145.6662ā1000
1
ā
45145.67
A
1
ā
ā45145.67
Month 2:
2
=
(
45145.67
Ć
1.0031667
)
ā
1000
A
2
ā
=(45145.67Ć1.0031667)ā1000
2
=
45288.1275
ā
1000
A
2
ā
=45288.1275ā1000
2
ā
44288.13
A
2
ā
ā44288.13
Month 3:
3
=
(
44288.13
Ć
1.0031667
)
ā
1000
A
3
ā
=(44288.13Ć1.0031667)ā1000
3
=
44427.4715
ā
1000
A
3
ā
=44427.4715ā1000
3
ā
43427.47
A
3
ā
ā43427.47
Month 4:
4
=
(
43427.47
Ć
1.0031667
)
ā
1000
A
4
ā
=(43427.47Ć1.0031667)ā1000
4
=
43563.6902
ā
1000
A
4
ā
=43563.6902ā1000
4
ā
42563.69
A
4
ā
ā42563.69
Month 5:
5
=
(
42563.69
Ć
1.0031667
)
ā
1000
A
5
ā
=(42563.69Ć1.0031667)ā1000
5
=
42696.7823
ā
1000
A
5
ā
=42696.7823ā1000
5
ā
41696.78
A
5
ā
ā41696.78
Month 6:
6
=
(
41696.78
Ć
1.0031667
)
ā
1000
A
6
ā
=(41696.78Ć1.0031667)ā1000
6
=
41826.7444
ā
1000
A
6
ā
=41826.7444ā1000
6
ā
40826.74
A
6
ā
ā40826.74
Month 7:
7
=
(
40826.74
Ć
1.0031667
)
ā
1000
A
7
ā
=(40826.74Ć1.0031667)ā1000
7
=
40953.5722
ā
1000
A
7
ā
=40953.5722ā1000
7
ā
39953.57
A
7
ā
ā39953.57
Month 8:
8
=
(
39953.57
Ć
1.0031667
)
ā
1000
A
8
ā
=(39953.57Ć1.0031667)ā1000
8
=
40077.2624
ā
1000
A
8
ā
=40077.2624ā1000
8
ā
39077.26
A
8
ā
ā39077.26
Month 9:
9
=
(
39077.26
Ć
1.0031667
)
ā
1000
A
9
ā
=(39077.26Ć1.0031667)ā1000
9
=
39197.8117
ā
1000
A
9
ā
=39197.8117ā1000
9
ā
38197.81
A
9
ā
ā38197.81
Month 10:
10
=
(
38197.81
Ć
1.0031667
)
ā
1000
A
10
ā
=(38197.81Ć1.0031667)ā1000
10
=
38315.2176
ā
1000
A
10
ā
=38315.2176ā1000
10
ā
37315.22
A
10
ā
ā37315.22
Month 11:
11
=
(
37315.22
Ć
1.0031667
)
ā
1000
A
11
ā
=(37315.22Ć1.0031667)ā1000
11
=
37429.4764
ā
1000
A
11
ā
=37429.4764ā1000
11
ā
36429.48
A
11
ā
ā36429.48
Month 12:
12
=
(
36429.48
Ć
1.0031667
)
ā
1000
A
12
ā
=(36429.48Ć1.0031667)ā1000
12
=
36540.5857
ā
1000
A
12
ā
=36540.5857ā1000
12
ā
35540.59
A
12
ā
ā35540.59
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
Ć
12
=
12000
1000Ć12=12000
Total amount at the end of the year:
12
ā
35540.59
A
12
ā
ā35540.59
Total interest earned:
Interest
=
12
ā
(
46000
ā
12000
)
Interest=A
12
ā
ā(46000ā12000)
Interest
=
35540.59
ā
34000
Interest=35540.59ā34000
Interest
=
1554.59
Interest=1554.59
The closest answer to $1554.59 is:
Answer:
A $1568
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