C; = i Co. Co = C t - I. C; - Co 1 C - Co = (1 + iy. t. Co (et ye. i. c; log(l + i) C, = Co(l + Lt) C; = C o (l + i)t. C t. o-l+i.

Transcription

1 D' Arcy McGee High School I May 26, 2017 est 11 Math Total Competency 2 Test: Mathematics of Finance & Probability Duration: 50 minutes 82 Name: 50WT/oflis Section MCU504: Useful Mathematics of Finance Formulas Investment with annual simple interest rate Investment with annual interest rate compounded annually C, = Co(l + Lt) C; = C o (l + i)t C - o-l+i.t i= Ct C t Co = (1 + iy C; - Co 1 t. Co (et ye i Co C t - Co log(c t -7- Co) t= t= i. c; log(l + i) I = Co. i. t 1= C; - Co -.' i I = C o (l + i)t - I I I - -- t=- Co - - C; = Co t Co.i Lt I + Co I l = (Ct-/) t Co = C t - I Co I t= (Ct-I).i... Co($): actual value or the initial value of the capital i (%); annual simple interest rate or annual interested compounded annually, depending on the type of investment t (years): the investment time in years 1 ($): accumulated interest value received at the end oftime t Cd$): the future value of the investment after time t years MCU504:C2 test/mathematics of Finance I Exponential & Log Numbers I Capital Investments and Probability Page 1

2 Part A: Multiple Choice Questions. Each question is worth 4 marks: Circle your answer. 1. A population of insects doubles every 3 days. If there are currently 600 insects, which of the equations below will determine the population P of insects at time t? t is the number of days. t A) P = 600 (~)3 B) P = 600 (;)3t C) P = 600(2)3t D) P = 600(2)3 t 2. A jar contains 2 green marbles and 3 blues marbles. A second jar contains 4 blue marbles and 2 green marbles. The experiment consists of drawing two marbles, one from each jar. What are the odds against getting two blue marbles? p(plu ~N1> fjjlu6) = 2- X.!:t ==- ~ ~ 2., A) 7:4 C) 2:3 3. Given: 2250 = (1.12)X, which ofthe equations below will solve for x?.s ~. 3D ij C B) 4:7 ] " If. odls r:., Blw/~ Afjsw&-R ~ D) 3:2 i Sf f): 3 A) log1.12 X = 2250 B) log, 1.12 = 2250 C) IIogu x, ")~. D) log, 2250 = 1.12 'e,y 1) =ffn IT,OrJ ANS'w~, odd ~ If ~;i1sri( f) lb~{~ " r s 3~2-4. A capital of $5000, invested at an annual interest rate of 2% compounded annually, generates at the end of the investment a capital equal to $7200. Which of one of the equations below will solve for the time period of the investment t (years)? A) log(7200-;- 5000) t = ---'::...: '- log(3) B) log( ) t = ---''--' '- log(1.02) C) log(7200 -;-5000) t = ---'::...: '- log(1.02) D) == MCU504:C2 test/mathematics of Finance I Exponential & Log Numbers I Capital Investments and Probability Page Z

3 Part B: Short Answer Questions Each right answer is worth 4 marks. A mark of zero is attributed to a wrong answer regardless of work shown.' 5. The depreciation of the value for a motorcycle is modeled by y = 22,500(.88) x for x years since \ In what year was the value of the car was $5500? --;i>" 65bD.::: 2 2 ~od{d'&1?)~ y..= If!} rs-~fll,)o{)j,,~ 1Ij'.RttiS ~ 7l-v.!htt' e 200:r+-l/; 2020,,' logd.kg The year the value of the car is worth $5500 is: ~ 2-020J ~ 6. A town of7125 inhabitants grows at a rate of3.1 % every year. Find the size of that town in 10 years. The size of that town in 10 years is: [96"68 ] --, ~ j ~~::; 7/?5'(J+ 0,(3/)'::::: Q StJLj 7. A capital of $2800, invested at an annual interest rate compounded annually, generates an accumulated capital equal to $5200 after 8 years. What was the annual interest rate compounded annually? i-:(t)~- i=rr::o~,)y8-j J -7 == O.fJ801aZ []?t- I)r J-- ~~ ~ The annual interest rate compounjed annually was 8. The Venn diagram below represents the distribution of 90 students according to courses they are currently taking. Distribution of the students according to courses they registered in Arabic (31) Spanish ( -3 b) a) Given that the student chosen is studying French, what is the probability that he/she is studying ArAnabic? q~io or (),22'; or 22. S-~ ~ swer: /4 ~ b) Given that the student chosen is studying Spanish, what is the probability that he/she is studying French? > I. 6 d) Answe;: /3' Dr {),16666., or I,7/; ITDJ MCU504:C2 test/mathematics of Finance I Exponential & Log Numbers I Capital Investments and Probability, Page 3

4 Part C: Long Answer Questions. Each uestion is worth 10 marks. Show all your work. 9. An advertising company employs 210 people. There are 125 full-time-employees and the rest are parttime employees. There are 140 male employees. There is an equal number of full-time and part-time female employees. A male employee is selected at random to be interviewed for a new advertising campaign. Given that a man is chosen, what is the probability that he is a full-time employee? S OlUTJ'otV :,. tull-,.,me ) faa.- TInE ~OIALeFT r r p.r> f:-emale (r:) '3S- 3S- 70. HA LE (I-/f ) CJo IO~AL. /ZS- 8S- 210 ~() pr)~ I zs-.- ~ ~ :;. 9 D ~ P( F'T M) ::: C ~PT=2Io- 12~=8f -:If.F =- lid - '140= 70 :#fo F-rJ==- :tt(h1 PT) ~ 1!2 - ]t2~3~ L 2- *0 P1):::. ~5-:,S- ::::.5"0 MCU504:C2 test/mathematics of Finance I Exponential & Log Numbers I Capital Investments and Probability Page 4

5 10. Caroline invests, for 5 years, a capital at an annual interest rate of 6%compounded annually. At the end of the investment, she invests the acquired value at a simple annual interest rate of 4% over the next two years. If the value acquired after7 years is equal to $ , what was Caroline's initial investment capital? Caroline's initial investment capital was: f-~---- ~~03C;~~ MCU504:C2 test/mathematics of Finance I Exponential & Log Numbers I Capital Investments and Probability Page 5

6 11. The following is some information on two games of chance Gamel After placing a bet, the player rolls a die If you roll an even number, the player will receive $12 If you roll an odd number, the player will receive $2 Game 1 is fair. Game 2 After placing a bet, the player randomly draws a marble from a bag The bag contains 50 congruent marbles: 11 are red, 15 are yellow and 24 are green. If a red marble is drawn, the player will receive $20 If a yellow marble is drawn, the player will receive $ 5$ If a green marble is drawn, the player will receive $2 A player must bet the same amount of money to play either Game 2 or Game 1. Is Game 2 to the player's advantage, to the player's disadvantage, or is it a fair game? Explain your answer. $OLU 11~tJ ~ t1)v-. S-O-L-' V:-F--Ft.-l5R-6Rn--E---:"j-F-'---'I2-S-1-Tt-D-P-(;-'"-7-E-I2{7---'-1 ;J.-'E-_-T-'If -"""='"B-: ~T=-t /.Q ~ X b e. the b.et (~ do Ilacs ~Q... / /5.pu.,...-i' - i V. ~ e.v.=f(eife/l)4q 2- x).+ P{odJ)e(2-f.); 0 -? e,\). = C).S(12-~'i- O.S-l2- y..)~d -r 0,1; (rz).; 0.5(>1) -+ t9,'s-(2)+ O.5{-'I) z: o o o.~~ z: 0 tj -1 ty;: 1]- 7-'X- [X==7 \ :::::::0,~Ti..e bel it red fvlj r~l~ h, MCU504:C2 test/mathematics of Finance / Exponential & Log Numbers / Capital Investments and Probability Page 6

7 P{R-Efl)=- Jl ; P(Ye"W>vJ)~ if ; P(Gt:EEN)=- ~ S6 (20-7),+~(S-7)f?!L(2-7) 5D so S-O -7 etv~==.g.o(13) + E(2)+ ~(-5) v 50 st) f43-3u. - /20-50'+ SO -t-. ~ D -/2D so D 50 -? E, 'V~ - 7,LUh;ch /'s Cl nrja-h~e value.. SD. ~V. <0 ;> 7h,'s i/llp/'fs, ~IY1R2 /5 fo [hp (-> / a,!12r 15 cu' s Q. r:1-ja n 'G..(j e., Your answer: select one of the checkboxes below: D Game 2 is fair : ~ is to the player's advantage Game 2 is to the player's disadvantage.«a- N.st» E: R-. MCU504:C2 test/mathematics of Finance I Exponential & Log Numbers I Capital Investments and Probability Page 7

8 12. Calissa is planning to invest her money at the bank:. She is tom between two bank institutions. Bank 1 A personal banker told Calissa to invest at Bank! with an annual simple interest rate of 6%. After crunching few numbers, the banker showed her the table below. Her money will grow up to $ after t years, and her accumulated interest (I) will be $ Co ($) i t (years) I C t / / % 5.07 $ $ Co($): the actual value or the initial value of the capital i (%): annual simple interest rate t (years): the investment time in years ] ($): accumulated interest value received at the end of time t Cd$): the future value of the investment after time t years Determine Calissa's initial capital and the time period t (years) of her investment? Bank 2 Investing the same initial capital for the same number of years of investment found in Bank 1, a personal banker at Bank 2 offered Calissa an annual interest rate compounded annually of 5.5% Co ($) i t (years) /($) C t ($) / Ir;-O. "71 5.5% ~~ 150~~ C o ($): the actual value or the initial value of the capital i (%): annual interest rate compounded annually t (years): the investment time in years ] ($): accumulated interest value received at the end of time t Cd$): the future value of the investment after time t years With which bank would you advise Calissa to invest her money? Round all numbers to the nearest hundreth. Justify your answer by filling the missing information in both tables above first, then write your conclusion. MCU504:C2 test/mathematics of Finance I Exponential & Log Numbers I Capital Investments and Probability Page 8

9 MCU504:C2 test/mathematics of Finance I Exponential & Log Numbers I Capital Investments and Probability Page 9

10 13. Managing deer population has been a real challenge in Canada. In some provinces and territories the deer population increases while in other regions it decreases at an alarming rate. Based on the 2010 statistics data, the deer population in the province of New Brunswick was at 78,530 and it increases at a rate of 1.5% every year; where as in Ontario, the deer population decreases by 0.85% every year. The deer population in Ontario that was recorded in 2010 was at 431,915. If this trend continues, when will the deer populations of both provinces, New Brunswick and Ontario, be the same? 't;#~ ~:=rr:":, jj.tw.- 8rulIs(}J/cJ(, ~ Olt-W;O_ ~ Y i t = _~X 79 1 S3D (/+D.OI,)).J{=_q3/,1 1S-(t-{},OD85; # :cc 73t'S3D Cl, D 15) X 11 -z: I; 3/"'lIS- (a, 91 1 [;) X r!;7 div;dt~8- ~i7& S-3o tal ~fd~s t": +Wo r,dijlnlej ;VJi/l be same: ~. -- y -:;::. 'I )t 78 S 30 (t. 0I'0./ =: Lt 3/. '1/')(0,~q Is) Ig53 0 c: 01 s) 1--:;:: 1.(3( q Ire0 qq Is))( 7<6 S?o' 78 ~">;O _ x (I. 0 I ';)) ;<:::=:, S.5 ( 0 ' q q I ') ) c,.0/5)< _ S-'~cP,qqlS)" - ID ~ ':) ~ (J'O/~ ),c.= 5.'5"" ~ ~=- ~ {),qqjs lo~ (I. D/~-7 o"q9/r-) y:: i.eaf$ z: The year the deer populations of both provinces will be the same is uo(52... ~o ~~ ;"5 cjb ac~ff..ed MCU504:C2 test/mathematics of Finance I Exponential & Log Numbers I Capital Investments and Probability Page 10