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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%
=
12
0.038
=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
.
Initial amount:
0
=
46000
A
0
=46000
For each month
n (from 1 to 12):
=
(
−
1
×
(
1
+
)
)
−
1000
A
n
=(A
n−1
×(1+r))−1000
We'll calculate this iteratively for each month.
Month 1:
1
=
(
46000
×
(
1
+
0.0031667
)
)
−
1000
A
1
=(46000×(1+0.0031667))−1000
1
=
(
46000
×
1.0031667
)
−
1000
A
1
=(46000×1.0031667)−1000
1
=
46145.6662
−
1000
A
1
=46145.6662−1000
1
≈
45145.67
A
1
≈45145.67
Month 2:
2
=
(
45145.67
×
1.0031667
)
−
1000
A
2
=(45145.67×1.0031667)−1000
2
=
45288.1275
−
1000
A
2
=45288.1275−1000
2
≈
44288.13
A
2
≈44288.13
Month 3:
3
=
(
44288.13
×
1.0031667
)
−
1000
A
3
=(44288.13×1.0031667)−1000
3
=
44427.4715
−
1000
A
3
=44427.4715−1000
3
≈
43427.47
A
3
≈43427.47
Month 4:
4
=
(
43427.47
×
1.0031667
)
−
1000
A
4
=(43427.47×1.0031667)−1000
4
=
43563.6902
−
1000
A
4
=43563.6902−1000
4
≈
42563.69
A
4
≈42563.69
Month 5:
5
=
(
42563.69
×
1.0031667
)
−
1000
A
5
=(42563.69×1.0031667)−1000
5
=
42696.7823
−
1000
A
5
=42696.7823−1000
5
≈
41696.78
A
5
≈41696.78
Month 6:
6
=
(
41696.78
×
1.0031667
)
−
1000
A
6
=(41696.78×1.0031667)−1000
6
=
41826.7444
−
1000
A
6
=41826.7444−1000
6
≈
40826.74
A
6
≈40826.74
Month 7:
7
=
(
40826.74
×
1.0031667
)
−
1000
A
7
=(40826.74×1.0031667)−1000
7
=
40953.5722
−
1000
A
7
=40953.5722−1000
7
≈
39953.57
A
7
≈39953.57
Month 8:
8
=
(
39953.57
×
1.0031667
)
−
1000
A
8
=(39953.57×1.0031667)−1000
8
=
40077.2624
−
1000
A
8
=40077.2624−1000
8
≈
39077.26
A
8
≈39077.26
Month 9:
9
=
(
39077.26
×
1.0031667
)
−
1000
A
9
=(39077.26×1.0031667)−1000
9
=
39197.8117
−
1000
A
9
=39197.8117−1000
9
≈
38197.81
A
9
≈38197.81
Month 10:
10
=
(
38197.81
×
1.0031667
)
−
1000
A
10
=(38197.81×1.0031667)−1000
10
=
38315.2176
−
1000
A
10
=38315.2176−1000
10
≈
37315.22
A
10
≈37315.22
Month 11:
11
=
(
37315.22
×
1.0031667
)
−
1000
A
11
=(37315.22×1.0031667)−1000
11
=
37429.4764
−
1000
A
11
=37429.4764−1000
11
≈
36429.48
A
11
≈36429.48
Month 12:
12
=
(
36429.48
×
1.0031667
)
−
1000
A
12
=(36429.48×1.0031667)−1000
12
=
36540.5857
−
1000
A
12
=36540.5857−1000
12
≈
35540.59
A
12
≈35540.59
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
×
12
=
12000
1000×12=12000
Total amount at the end of the year:
12
≈
35540.59
A
12
≈35540.59
Total interest earned:
Interest
=
12
−
(
46000
−
12000
)
Interest=A
12
−(46000−12000)
Interest
=
35540.59
−
34000
Interest=35540.59−34000
Interest
=
1554.59
Interest=1554.59
The closest answer to $1554.59 is:
Answer:
A $1568
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