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# FINITE MATH & ITS APPLIC MA 162

UK

GPA 3.54

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This 7 page Class Notes was uploaded by Kennith Herman on Friday October 23, 2015. The Class Notes belongs to MA 162 at University of Kentucky taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/228160/ma-162-university-of-kentucky in Mathematics (M) at University of Kentucky.

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Date Created: 10/23/15

DEPARTMENT OF MATHEMATICS Problems from old third exams Useful formulas o 1Simple Interest I PTt Accumulation A Pl rt 0 2Compound lnterest Accumulation A Pl Present value P A1 i o 3Effective rate Teff 1 71 Annuity 41Sum S 7 RW 4 Present value P 0 Set counting 5Two sets nAU B nA nB 7 nA QB 6 Three sets nAUBUC nA nB nC 7 nA B 7 nB C 7nC AnA B C 1 Suppose that A B and C are sets with 93 69 and 86 members respectively Calculate the indicated quantities Display correct formulas or appropriate Venn diagrams i If A B has 45 members then AU B has members Answer Using the two set counting formula 5 we get nAUB 7 74A 7 nB 7 nA B 7 93 69 7 45 7117 ii If it is further known that A C has 56 members then AUC has members Answer Using the two set counting formula5 we get nAU 0 7 nA 7 nC 7 Wm 0 7 93 86 7 56 7123 iii If in addition B 7 C has 39 members then B H C has l members Answer Note that by the same formula we get nB nB 7 C nB H C since B 7 C and B H C have empty intersection Hence 69 39 nB H C so nB H C 69 7 39 30 iv Finally if we are given that the intersection of all three sets A B and C has 16 members then the union of all three sets has D members N 9 9 Answer Use the three set counting formula 6 and the above results to get nAUBUC 9369867 45563016 133 Bill invests 4400 at 7 compounded quarterly a After 10 years his investment will grow to D Answer Since this is a single investment compounded we use formula 2 We have 7 007 m 4 so i 0074 Also t 10 so n 104 40 Also P 4400 So the answer is 44001 007440 880703 b His investment will have grown to 22000 after 1 years Answer The same formula now gives 22000 44001 0074 where the new number t of years is being asked Taking logs and simplifying 4tlog1 0074 log220004400 and this gives 4t 927703 or t 2319 Jack needs 10 000 after 10 years from now How much would he have to invest now at 7 interest compounded annu ally to have that sum available Be sure to show the formula used Answer This is still a single investment with compound interest so we use formula 2 and note that i 0071 n 1 10 10 and A 10000 So 10000 P1 00710 or P 100001 007 10 508349 You plan to pay for the purchase of a house with a 15 year loan of 120 thousand dollars at 66 interest compounded monthly What is your monthly payment of principal and interest and what is the your total payout Monthly Payment 1 dollars Be sure to show the formula used 539 533 Answer This is a monthly payment situation so we need formula 41 or 42 Also we are given P the present value so use 42 Using the values 7 66100 m 12 t 15 P 120000 we get 239 661200 n 12 15 180 and the equation 120000 R1 7 1 061200 18 661200 Evaluating we get 120000 R1140752759 or R 1200001140752759 Total payout D dollars This does not need any formula We pay the above amount R monthly for 15 years so the payout is 10519412 15 18934865 Suppose that you decide to save up for a car purchase in 5 years You want to have 14000 saved and you will make monthly payments into an account paying 8 interest compounded monthly Be sure to show the formulas used a How much should your monthly payment be Answer This is again an annuity case since we are making periodic payments This time the future value 14000 is given so use 41 1257 14000Rlt181200 1gt 81200 or 14000 R 73147686070 or R 1400073147686070 19015361752 so you report 19054 as the monthly payment b If at the end of 5 years you decide to keep the old car another year and continue making the same payments how much additional money will accumulate into the account Answer The net accumulation after 6 years at the same payment rate would be 1753450654 or 1753451 so we would have an additional 353451 dollars extra over the original sum of 14000 In this problem assume a 360 day year 77Rent a Check77 will loan you 55 of your paycheck of 300 for 7 days After 7 days you pay back the loan plus 10 a Calculate the annual simple interest rate for this service Answer 1 per cent Surprised Now you know what a loan shark isll Answer This is a simple interest rate calculation using formula 1 We have t 7360 P 300055 and I 10 so I 10 7 3116883117 T Pt 1657360 F1 9 So the rate is 311i69i b Across the street 77The Loan Shop77 will loan you 5 more of your paycheck and charges you an interest of 11 Is this a lower rate Explain your answer by comparing the rates Be sure to show all work Answer This is the same problem with different P and 1 Thus the new rate is 1 i 3142857143 T Pt 1807360 which is 3142996 or even higher Sam has a library of different but colorful books He has 6 red books 13 blue books and 4 yellow books a Suppose he lets a friend take home three books one of each color How many possiblities are there 7 Answer Answer By the multiplication principle we get the answer as 6 13 4 b Suppose instead he decides his friend can only borrow two books and they must be the same color How many possibilities now Answer There are three separate cases depending on the color chosen The friend can choose 2 red books in C62 525 1312 428 2 78 ways and yellow books in The total count is 15 78 6 99 Note that the nal numbers are not multiplied but simply added We cannot choose different colors simulta neouslyl 15 ways blue books in 6 ways There are 12 members in an executive committee A slate of three mem bers is to be nominated from them to work as of cers the President the Secretary and the Treasurer a How many different slates can be formed from the executive committee Answer Answer lmagine setting up chairs with titles on themi The president s chair can be lled in 12 ways Now only 11 members are available for the secretary s chair and then only 10 left for the treasurer7s chair Since these choices have to be made at the same time to make a slate the answer is 12 ll10 1320 We could also explain this by the formula P123 l2 ll10i b Suppose that Mr Smith is already the chosen nominee to become the Secretary In that case how many ways are there to complete the slate 5 10 Answer Answer Now remove Mr Smith from available candidate list and note that only two more positions have to be lled So the answer is P11 2 1110 110 Jack and Lisa are getting married and hoping to buy a small house worth 60 000 They need to come up with 12 000 for down payment and have only 2 000 saved for this a a Fortunately their parents offer them a simple interest loan for the remaining 10 000 at a very generous 45 APR annual percentage rate for a period of 5 years If they accept this offer how much net amount would they pay back to their parents after the 5 years Answer This is formula 1 applied A 100001 Airs1005 12250 The interest expense is thus 2250 After discussion with the mortgage company the aggressive sales manager said that he can do better and offered them an APR of 411 compounded annually for the same period and the same amount By the way77 he added casually that there is a small processing fee of only 40 77 Should Jack and Lisa go for the parents7 offer or the manager s offer Justify your answer with solid numbersl Answer This time the net expense is interest plus 40 but we need to use the compound interest formu a 2 10000 1 411005 40 12225113 40 12265113 Thus they lose more with the bank If the manager were to offer an APR of 4 but now compounded monthly and with the same processing fee what should Jack and Lisa do Again justify your answers with precise calculated numbers Answer This time our accumulation comes out 12209197 Even with 40 fee this is 122499 or 3 cents cheaper than the parents 1f the manager offers Jack and Lisa an APR of 4 compounded monthly for a period of 15 years provided they borrow 70000 what would their monthly payment be Answer We use formula 42 to write 7 17 1 41200lt12gtlt15gt 70000 7 R W Solution gives R 517178 b Jack and Lisa now do some calculations and gure that they can afford between 500 and 550 in monthly payments If they can get an APR of 5 compounded monthly and want to pay off the loan in 10 years what are the least and most amounts they can hope to borrow Answer We use R 500 and 550 respectively with 7 0 05 and t 10m 12 Then the values of P from the same formula give 47140168 and 51854i75 respectively These are the limits for the 10 year repaymenti c Jack and Lisa want to get new furniture but decide that they want no more loans and they can wait for two years before they get the furniture that they love They decide to start a savings account which pays an APR of 315 compounded monthly and decide to put 250 into it every month How much can they afford to pay for the furniture after the two years Answer Here the future value is desired with R 250 7 315100 m 12t 2 Thus the formula 41 gives the answer 620562 11 Answer the following a In a survey of 1250 likely voters in Fayette county 670 said that they would vote for the mass transit proposal and 875 said they would vote for the marriage amendment 1f 50 were not going to vote for either of the two amendments how many were going to vote for both the amendments Answer Let A and B be the sets of people voting for the mass transit and Marriage amendment respective y We know nA 670 nB 8751 Clearly nAU B 1250 7 50 1200 Thus by formula 5 1200 670 875 7 nA B so the desired answer is 345 In the same survey how many were going to vote for only one of the two amendments Answer This asks you to add up nA 7 A H B and nB 7 A H B and these are clearly 670 7 345 875 7 345 respectively The answer is 325 530 855 This can also be calculated as nAU B 7 nA H B b In a survey of 400 students it was found that 220 usually went to the student center cafeteria to eat 190 often used some other nearby campus cafeteria while 70 claimed that they go to both regularly The rest reported that they don7t eat on campus How many don7t eat on campus Answer Let A be the set of center cafeteria clients and B be the set of other cafeteria students Then nA U B is the number of students eating on campus and the rest from 400 eat elsewhere We see from formula 5 nA UB 220 190 7 70 340 So the left over 400 7 340 60 eat away from the campus In a survey 100 voters were asked which of the three election issues terrorism economy and morality was important for them They could vote for more than one choice if desired Of these 10 declared that every issue was very important for them There were 40 voters who considered that the the terrorism issue was important 26 thought that the economy was important and 65 thought that the morality issue was very important 0 Determine the number of voters who found that two or more issues were important for them 0 Determine the number of single issue voters ie voters who considered only one of the three issues to be important Answer Let AB C be the sets of voters who consider terrorism economy and morality important respectively Here is what we know 0 nAUBUC 100 nA B C 10 o nA 40nB 26 nC 65 0 From formula 6 we deduce that 100 4026657 ltnA B nB o nC Agt 10 Thus nA BnB CnC A 402665107100 41 o The set voters voting for two or more issues is the union of the three sets A Q B B Q C C Q A We note that the intersection of any two of these is always the same set A H B H C which has 10 element s 0 Now formula 6 applied to these three sets gives Wm B Una 0 Uc A 41710101010 21 This is the number of people voting for at least two issues 0 Finally single issue voters are the remaining 100 7 21 79

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