```8 AM Class Group 3

Math Project Group 3 Part 1:  Chase Chemero, Daniel Hanley, Sonja Henst,
Luis Macancela, Maxim Shorey, Thanh Tran

Formula Used:
Amount in savings
not including interest  +(( Annuity amount + Interest earned) ˆ (( Annuity
amount + Interest earned) * Tax percentage))) = New
earned because must         per year                 last year
per year                   last yea
Savings amount
be taxed

New savings amount + (New Savings amount * Interest percentage) = End of
year total

Justification for formula used:  I used the above formula to get my answer
because as stated in the instructions you will receive for 20 years a
\$50000  annuity and receive and interest of 6 % for what has been invest
also at the beginning of each year you will be taxed 30 % on not only the
annuity payment but also the interest you have accumulated over the past
year.
So as each step is followed (with the exception to the first) the interest
earned over the year is taken out of the savings added to the annuity and
then taxed, followed by being added back into saving to be reinvested for
a years more of interest this is repeated for 20 years until the 21rst
year where you receive no annuity payment and only earn and taxed on your
interest as stated in the instructions.

Work
50000-(50000*.3)=35000
35000*.3=2100
35000+2100 =                      37100 End year 1 total

35000+((50000+2100)-((50000+2100)*.3)))=71470
52100 -   15630 =36470
71470+(71470*.06)=75758.20  End year 2 total
4288.2

71470+((50000+4288.2)-((50000+4288.2)*.3)))=109471.74
54288.2  -    16286.46 = 38001.74
109471.74+(109471.74*.06)=116040.04 End  year 3 total
6568.30

109471.74+((50000+6568.30)-((50000+6568.30)*.3)))=149069.55
56568.30          - 16970.49 = 39597.81
149069.55+(149069.55*.06)= 158013.72  End year 4 total
8944.17

149069.55+((50000+8944.17)-((50000+8944.17)*.3)))= 190330.47
58944.17         -   17683.25  = 41260.92
190330.47+(190330.47*.06)= 201750.30   End year 5 total
11419.83

190330.47+((50000+11419.83)-((50000+11419.83)*.3)))=233324.35
61419.83            - 18425.95  =  42993.88
233324.35+(233324.35*.06)= 247323.81  End year 6 total
13999.46

233324.35+((50000+1399.46)-((50000+13999.46)*.3)))= 278123.97
6399.46           -  19199.84  = 44799.62
278123.97+(278123.97*.06) = 294811.41   End year 7 total
16687.44

278123.97+((50000+16687.44)-((50000+16687.44)*.3))) = 324805.18
66687.44            -  20006.23 =  46681.21
324805.18+(324805.18*.06) = 344293.49  End year 8 total
19488.31

324805.18+((50000+19488.31)-((5000+19488.31)*.3))) = 373447
69488.31           -  20846.49  = 48641.82
373447+(373447*.06) = 395853.82   End year 9 total
22406.82

373447+((50000+22406.82)-((50000+22406.82)*.3))) = 424131.77
72406.82      -  21722.05   = 50684.77
424131.77+(424131.77*.06) =449579.68 End year 10 total
25447.91

424131.77+((50000+25447.91)-((50000+25447.91)*.3))) = 476945.31
75447.91          -  22634.37  = 52813.54
476945.31+(476945.31*.06) = 505562.03  End year 11 total
28616.72

476945.31+((5000+28616.72)-((50000+28616.72)*.3))) =531977.01
78616.72         -  23585.02  =  55031.70
531977.01+(531977.01*.06) = 563895.63  End year 12 total
31918.62

531977.01+((50000+31919.62)-((50000+31918.62)*.3)))= 589320.04
81918.62          -  24575.59  =  57343.03
589320.04 + (589320.04*.06) = 624679.24  End year 13 total
35359.20

589320.04 +((50000+35359.20)-((50000+35359.20)*.3))) = 649071.48
65359.20            -   25607.76 = 59751.44
649071.48+(649071.48*.06) = 688015.77  End year 14 total
38944.29

649071.48+((50000+38944.29)-((50000+38944.29)*.3))) = 711332.48
88944.29           -  26683.29 =  62261
711332.48+(711332.48*.06) = 754012.43  End year 15 total
42679.95

711332.48+((50000+42679.95)-((50000+42679.95)*.3))) = 776208.44
92679.95           -   27803.99  = 64875.96
776208.44+(776208.44*.06)  = 822780.95 End year 16 total
46572.51

776208.44+((50000+46572.51) ˆ((50000+46572.51)*.3))) = 843809.20
96572.51            -  28971.75 = 67600.76
843809.20+(843809.20*.06) = 894437.75  End year 17 total
50628.55

843809.20+((50000+50628.55)-((50000+50628.55)*.3))) 914249.18
100628.55         -  30188.57 = 70439.98
914249.18+(914249.18*.06) = 969104.13  End year 18 total
54854.95

914249.18+((50000+54854.95)-((50000+5485.95)*.3))) = 987647.64
104854.95          -  31456.49  = 73398.46
987647.64+( 987647.64*.06) = 1046906.50  End year 19 total
54258.86

987647.64+((50000+59258.86)- ((50000+59258.86)*.3))) = 1064128.84
109258.86         -      3277.66   = 76481.20
1064128.84+(1064128.84*.06) = 1127976.57 End year 20  total
63847.73

1064128.84+(63847.73-(63847.73*.3))= 1108822.25
63847.73-19154.32 =44693.41
1108822.25+(1108822*.06)= 1175351.59 End year 21
66529.34

Part 2

(Amount in savings + (interest amount       - (Interest amount    * .03))
= New savings Amount
with out interest)	 from previous year) from previous year)

New savings amount+( New savings amount *.06 ) = End of year total

Justification for formula used: I used this process for figuring out the
lump sum payment under the same conditions as part 1 because your only
income is the 1rst and only initial lump sum payment you must pay a huge
tax on it which is one in the first part and proceeding after that is that
as stated in the instructions after each year you get interest on what you
have in your savings account it is only until the first day of the next
year are you taxed on you interest ( your only other source of income and
there fore must be taxed) But you aren‚t taxed on your interest until the
first day of the new year, after which you get to keep the rest of your
interest for further investment to gather another years interest to be

Work for lump sum

1000000-(1000000*.3)=700000
300000
700000+(700000*.06)=742000 End year 1

700000+(42000-(42000*.3))=729400
42000-12600=29400
729400+(729400*.06)=773164 End year 2
43764

729400+(43764-(43764*.3))= 760034.8
43764-13129.2=30634.80
760034.80+(760034.80*.06)= 805636.89 End year 3
45602.09

760034.8+(45602.09-(45602.09*.3))= 791956.26
45602.09-13680.63=31921.46
791956.26+(791956.26*.06)= 839473.64 End year 4
47517.38

791956.26+(47517.38-(47517.38*.3))= 825218.43
47517.38-14255.21=33262.17
825218.43+(825218.43*.06)=874731.54 end year 5
49513.11

825218.43+(49513.11-(49513.11*.3))=859877.61
49513.11-14853.93=34659.11
859877.61+(859877.61*.06)=808284.95 end year 6
51592.66

859877+(51592.66-(51592.66*.3))= 895992.47
51592.66 ˆ 15477.8=36114.86
895992.47+(895992.47*.06) = 949752.02 End year 7
53759.55

895992.47+(53759.55-(5375955 *.3)) =933624.15
53759.55-16127.87=37631.68
933624.15+(933624.15*.06)=989641.60 End year 8
56017.45

933624.50+(56017.45-(56017.45*.3))= 972836.71
56017.45-16805.24= 39212.21
972836.71+(972836.71*.06)= 1031206.91 End year 9
58370.20

972836.71+(58370.20-(58370.20*.3))= 1013695.85
58370.20-17511.06=40859.14
1013695.85+(1013695.85*.06)= 1074517.60 End year 10
60821.75

1013695.85+(60821.75-(60821.75*.3)) = 1056271.08
60821.75-18246.53=42575.23
1056271.08+(1056271.08*.06)=1119647.34 End year 11
63376.26

1056271.08+(63376.26-(63376.26*.3)) = 1100634.46
63376.26-19012.88= 44363.38
1100634.46+(1100634.46*.06) = 1166672.53 End year 12
66038.07

1100634.46+(66038.07-(66038.07*.3)) = 1146861.11
66038.07- 19811.42=46226.65
1146861.11+(114681.11*.06) = 1215672.78 End year 13
68811.67

1146861.11+(68811.67-(68811.67*.3))= 1195029.28
68811.67- 20643.50= 48168.17
1195029.28+(1195029.28*.06) = 1266731.04 End year 14
71701.76

1195029.28+(71701.76-(71701.76*.3)) = 1245220.51
71701.76-21510.53=50191.23
1245220.51+(1245220.51*.06) = 1319933.74 End year 15
74713.23

1245220.51+(74713.23-(74713.23*.3))= 1297519.77
74713.23- 22413.97= 52299.26
1297519.77+(1297519.77*.06)= 1375370.96 End year 16
77851.18

1297519.77+(77851.19-(77851.19*.3))= 1382015.60
77851.16- 23355.36= 54495.83
1382015.60+(1382015.60*.06) = 14631636.54 End year 17
81120.94

1382015.60+(81120.94-(81120.94*.3)) = 1438800.26
81120.94-24336.28= 56784.66
1438800.26+(1438800.26*.06) = 1525128.28 End year 18
86328.01

1438800.26+(86328.01-(86328.01*.3))= 1499229.87
86328.01- 25898.40= 60429.61
1499229.87+( 1499229.87*.06)= 1589183.66 End year 19
89953.79

1499229.87+(89953.79-(89953.79*.3)) 1562197.52
89953.79-26986.14=62967.65
1562197.52+(1562197.52*.06) = 1655929.37 End year 20
93731.85
1562197.52+(93731.85-(93731.85*.3)= 1627809.81
93731.85-28119.56 = 65612.29
1627809.81+( 1627809.81*.06)= 1725478.40 End year 21
97668.59

Part 3

Formula

1000000                                X
------------                 =       --------

X:  Represents the answer for part 3 which is what the amount for a lump
sum payment that would leave the winner with the same amount as the
annuity after twenty years.

This is set up as proportion to be set up as

X =     (Part 1 answer) * (1000000)
--------------------------------------

This works because the \$1000000 relates to the Part 2 answers by \$1
million being a lump sum and the Part 2 answer by being a lump sum that is
being taxed with interest.
The Part 3 answer will relate to the Part 1 answer by what lump sum amount
will get you the same answer with taxes and interest as the Part 1 answer
for annuity payments.

With all the numbers put in it looks like

1000000                      X
------------            = -------------
1725478.40              1127976.57

In the correct format to solve for X is

X=   (1127976.57) * (1000000)
-------------------------------------
1725478.40

X= 653718.16

X is the lump sum payment that under the same conditions as Part 1 will
get you the same amount as Part 1

Part 4

Lump Sum: Pros

	Receive a larger amount of money at once
	Higher interest rates per year
	Greater sum in the end

Cons:

	Overall higher tax payments per year
	If incorporating a human-psychological factor: with so much money given
at once the lower the chances are for smart investments and savings
Annuity: Pros

	Overall lower tax payments per year
	If incorporating a human-psychological factor a steady increase in money
will increase the chances for smart investing and savings

Cons:
	Over all lower savings amount
	Less money overall per year and in the end
	Overall lower interest payments per year

Common Similarities that many be over looked by the normal person:

	With out incorporating taxes and interest, you essentially getting the
same amount of money whether it is \$50,000 for 20 years ( 50,000 * 20 =
700,000)  or \$700,000 at once

The only reason why the lump some payment is larger in the end over the
annuity ( regardless of the huge initial tax payment) is because the money
left over from the taxes is so large that the interest per year grows
faster than with the annuity payments thus resulting in a larger sum.

Part 5

There are multiple assumptions made in this project to make the problem
simpler to solve

The assumptions are as follows:

1.	There are no other expenditures ( such as money spent on housing,
clothes, food essentially any durable and non-durable goods were not
2.	There is no other source of income (whether he/ she has another job
with a income that can be added and taxed to this savings account other
possible winnings from other contests ect.)
3.	There are no other investments ( some people have many types of bank
accounts in this particular case there is just one savings account, some
people have checking account and other investment bank accounts)
4.	The interest earned each year is the same ( In reality savings interest
earned never stays the same nor is it ever that high)
5.	Taxed percentage stays the same ( In reality taxed percentage are
subject to change by either the states government and/or federal
government)
6.	There are only 2 taxes that have to be paid ( in reality there are many
other taxes that are paid besides just a federal and states tax, which can
include Medicare tax and Social security tax)
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