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Solving Coin and Attendance Problems, Exams of Discrete Mathematics

A collection of mathematical problems related to coins and attendance at various events. The problems involve determining the number of coins of each type based on their total value and the number of coins in the collection, as well as finding the number of adults and children who attended events based on the total number of attendees, admission prices for adults and children, and the total receipts.

Typology: Exams

2021/2022

Uploaded on 09/12/2022

dylanx
dylanx 🇺🇸

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4.5
1) AcollectionofdimesandquartersisworthS15.25.Thereare103coinsinall.Howmanyofeachis
there?
󰇛10󰇜󰇛
󰇜 󰇛103󰇜󰇛10󰇜
10 25 1525
10 10 1030

 

33 103 33
33 33
70 70 
33 
3) Theattendanceataschoolconcertwas578.Admissionwas$2.00foradultsand$1.50for
children.ThetotalreceiptswereS985.00.Howmanyadultsandhowmanychildrenattended?
2󰇛󰇜󰇛578󰇜󰇛2󰇜
2 1.5 985
2 2 1156
.
. 
.
342 578  342
342 342
236 236 
342 
5) Aboyhas$2.25innickelsanddimes.Iftherearetwiceasmanydimesasnickels,howmanyof
eachkindhashe?
5 20 225

 
  9 
918 
2
󰇛9󰇜18
7) Acollectionof27coinsconsistingofnickelsanddimesamountsto$2.25.Howmanycoinsofeach
kindarethere?
󰇛10󰇜󰇛
󰇜 󰇛27󰇜󰇛10󰇜
5 10 225
10 10 270

 

9
927
9 9 18 
18 9 
NVT
D1010D
Q2525Q
1031525
NVT
A22A
C1.51.5C
578985
NVT
N55N
D=2N1020N
225
NVT
NSSN
D1010D
27225
pf3
pf4
pf5

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  1. A collection of dimes and quarters is worth S15.25. There are 103 coins in all. How many of each is there? ሺെ10ሻሺ ܦ൅ܳ ሻ ൌ ሺ103ሻሺെ10ሻ 10 ܦ൅ 25 ܳൌ 1525 െ10 ܦെ 10 ܳൌ െ ଵହொ ଵହ ൌ^

ସଽହ ଵହ ܦ൅ 33 ൌ 103 ܳൌ 33 െ33 െ 33 ܦൌ 70 ݏ ݁݉݅݀ 70 ݏݎ݁ݐݎܽݑ ܳ 33

  1. The attendance at a school concert was 578. Admission was $2.00 for adults and $1.50 for children. The total receipts were S985.00. How many adults and how many children attended? െ2ሺ ܣ൅ ܥሻ ൌ ሺ578ሻሺെ2ሻ 2 ܣ൅ 1.5 ܥൌ 985 െ2 ܣെ 2 ܥൌ െ ି଴.ହ஼ି ଴.ହ ൌି^

ଵଵହ଺ି ଴.ହ ܣ൅ 342 ൌ 578 ܥൌ 342 െ342 െ 342 ܣൌ 236 ݏݐ݈ݑ݀ܣ 236 ݊݁ݎ݈݄݀݅ܥ 342

  1. A boy has $2.25 in nickels and dimes. If there are twice as many dimes as nickels, how many of each kind has he? 5 ܰ൅ 20 ܰൌ 225 ଶହே ଶହ ൌ^

ଶଶହ ଶହ ݏ ݈݁݇ܿ݅ܰ 9 ܰൌ 9 ݏ݁݉݅ܦ 18 ܦൌ 2ሺ9ሻ ൌ 18

  1. A collection of 27 coins consisting of nickels and dimes amounts to $2.25. How many coins of each kind are there? ሺെ10ሻሺ ܰ൅ ܦ ሻ ൌ ሺ27ሻሺെ10ሻ 5 ܰ൅ 10 ܦൌ 225 െ10 ܰെ 10 ܦൌ െ ିହேି ହ ൌି^

ସହି ହ ܰൌ 9 9 ൅ ܦൌ 27 െ9 െ 9 ݏ݁݉݅ܦ 18 ܦൌ 18 ݏ ݈݁݇ܿ݅ܰ 9

N V T

D 10 10D

Q 25 25Q

N V T

A 2 2A

C 1.5 1.5C

N V T

N 5 5N

D=2N 10 20N

N V T

N S SN

D 10 10D

  1. There were 429 people at a play. Admission was $1 each for adults and 75 cents each for children. The receipts were $372.50. How many children and how many adults attended? ሺെ1ሻሺ ܣ൅ ܥሻ ൌ ሺ429ሻሺെ1ሻ ܣ൅ .75 ܥൌ 372. ܥ െ ܣെ ൌ െ ି.ଶହ஼ି .ଶହ ൌି^

ହ଺.ହି .ଶହ ܣ൅ 226 ൌ 429 ܥൌ 226 െ226 െ 226 ܣൌ 203 ݏݐ݈ݑ݀ܣ 203 ݊݁ݎ݈݄݀݅ܥ 226

  1. There were 203 tickets sold for a volleyball game. For activity‐card holders, the price was $1. each and for non‐card holders the price was $2 each. The total amount of money collected was $310. How many of each type of ticket was sold? െ2ሺ ܣ൅ܰ ሻ ൌ ሺ203ሻሺെ2ሻ 1.25 ܣ൅ 2 ܰൌ 310 െ2 ܣെ 2 ܰൌ െ ି.଻ହ஺ି .଻ହ ൌି^

ଽ଺ି .଻ହ 128 ൅ ܰൌ 203 ܣൌ 128 െ128 െ 128 ܰൌ 75 ݀ݎܽܥ ݊݋ ܰ 75 ݀ݎܽܥ ݕݐ݅ݒ݅ݐܿܣ 128

  1. At a recent Vikings game $445 in admission tickets was taken in. The cost of a student ticket was $1.50 and the cost of a non‐student ticket was $2.50. A total of 232 tickets were sold. How many students and how many nonstudents attended the game? െ1.5ሺ5 ൅ܰ ሻ ൌ ሺ232ሻሺെ1.5ሻ 1.55 ൅ 2.5 ܰൌ 445 െ1.55 െ 1.5 ܰൌ 348 ܰൌ 97 ܵ൅ 97 ൌ 232 െ97 െ 97 ݏݐ݊݁݀ݑݐ ܵെ ݊݋ ܰ 97 ܵൌ 135 ݏݐ݊݁݀ݑݐ ܵ 135

N V T

A 1 A

C .75 .75C

N V T

A 1.25 1.25A

N 2 2N

N V T

N 2.5 2.5N

  1. A total of $27000 is invested, part of it at 12% and the rest at 13%. The total interest after one year is $3385. How much was invested at each rate? െ.12ሺ ݔ൅ ݕሻ ൌ ሺ27000ሻሺെ.12ሻ ݔ൅ 14500 ൌ 27000

. 12 ݔ൅ .13 ݕൌ 3385 െ14500 െ 14500 െ.12 ݔെ .12 ݕൌ െ3240 ݔൌ 12500 ି.଴ଵ௬ି .଴ଵ ൌି^

ଵସହି .଴ଵ ݕൌ 14500 $12,500 @12% $14,500 @ 13%

  1. A total of $9000 is invested, part of it at 10% and the rest at 12%. The total interest after one year is $1030. How much was invested at each rate? െ.1ሺ ݔ൅ ݕሻ ൌ ሺ9000ሻሺെ.1ሻ ݔ൅ 6500 ൌ 9000

. 1 ൅ .12 ݕൌ 1030 െ6500 െ 6500 െ.1 ݔെ .1 ݕൌ െ900 ݔൌ 2500 .଴ଶ௬ .଴ଶ ൌ^

ଵଷ଴ .଴ଶ ݕൌ 6500 $2500 @ 10% $6500 @ 12%

  1. An inheritance of $10000 is invested in 2 ways, part at 9.5% and the remainder at 11%. The combined annual interest was $1038.50. How much was invested at each rate? െ.095ሺ ݔ൅ ݕሻ ൌ ሺ10000ሻሺെ.095ሻ ݔ൅ 5900 ൌ 10000

. 095 ݔ൅ .11 ݕൌ 1038.50 െ5900 െ 5900 െ.095 ݔെ .095 ݕൌ െ950 ݔൌ 4100 ଴.଴ଵହ௬ .଴ଵହ ൌ଼଼^

.ହ .଴ଵହ ݕൌ 5900 $4100 @ 9.5% $5900 @11%

  1. Jason earned $256 interest last year on his investments. If $1600 was invested at a certain rate of return and $2400 was invested in a fund with a rate that was double the rate of the first fund, find the two rates of interest. 1600 ݔ൅ 4800 ݔൌ 256 ଺ସ଴଴௫ ଺ସ଴଴ ൌ^

ଶହ଺ ଺ସ଴଴ ݔൌ 0.04 $1600 @ 4% 2 ݔൌ 0.08 $2400 @ 8%

N V T

x .12 .12x y .13 .13y 27000 3385

N V T

x .10 .1x y .12 .12y 9000 1030

N V T

x .095 .095x y .11 .11y 10000 1038.

N V T

1600 x 1600x 2400 2x 4800x 256

  1. A total of $8500 is invested, part of it at 6% and the rest at 3.5%. The total interest after one year is $385. How much was invested at each rate? െ.035ሺ ݔ൅ ݕሻ ൌ ሺ8500ሻሺെ.035ሻ 3500 ൅ ݕൌ 8500 .06 ݔ൅ .035 ݕൌ 385 െ3500 െ 3500 െ.035 ݔെ .035 ݕൌ െ297.5 ݕൌ 5000 .଴ଶହ௫ .଴ଶହ ൌ଼^

଻.ହ .଴ଶହ ݔൌ 3500 $3500 @ 6% $5000 @ 3.5%

  1. A total of $15000 is invested, part of it at 8% and the rest at 11%. The total interest after one year is $1455. How much was invested at each rate? െ.08ሺ ݔ൅ ݕሻ ൌ ሺ15000ሻሺെ.08ሻ ݔ൅ 8500 ൌ 15000

. 08 ݔ൅ .11 ݕൌ 1455 െ8500 െ 8500 െ.08 ݔെ .08 ݕൌ െ1200 ݔൌ 6500 .଴ଷ௬ .଴ଷ ൌ^

ଶହହ .଴ଷ ݕൌ 8500 $6500 @ 8% $8500 @ 11%

  1. A total of $6000 is invested, part of it at 4.25% and the rest at 5.75%. The total interest after one year is $300. How much was invested at each rate? െ.0425ሺ ݔ൅ ݕሻ ൌ ሺ6000ሻሺെ.0425ሻ ݔ൅ 3000 ൌ 6000 .0425 ݔ൅ .0575 ݕൌ 300 െ3000 െ 3000 െ.0425 ݔെ .0425 ݕൌ െ255 ݔൌ 3000 .଴ଵହ௬ .଴ଵହ ൌ^

ସହ .଴ଵସ ݕൌ 3000 $3000 @ 4.25% $3000 @ 5.75%

  1. A total of $11000 is invested, part of it at 6.8% and the rest at 8.2%. The total interest after one year is $797. How much was invested at each rate? െ.068ሺ ݔ൅ ݕሻ ൌ ሺ11000ሻሺെ.068ሻ ݔ൅ 3500 ൌ 11000

. 068 ݔ൅ .082 ݕൌ 797 െ3500 െ 3500 െ.068 ݔെ .068 ݕൌ െ748 ݔൌ 7500 .଴ଵସ௬ .଴ଵସ ൌ^

ସଽ .଴ଵସ ݕൌ 3500 $7500 @ 6.8% $3500 @8.2%

N V T

x .06 .06x y .035 .035y 8500 385

N V T

x .08 .08x y .11 .11y 15000 1455

N V T

x .0425 .0425x y .0575 .0575y 6000 300

N V T

x .068 .068x y .082 .082y 11000 797