the one point contest - home - math · 2006. 8. 10. · help you familiarize yourself with ww. here...

U of U Math 4010-1 Summer 2006

Peter Alfeld.

WeBWorK assignment number 1.

due 5/31/06 at 11:59 PM.This is the first WeBWorK home work (ww hw)

set. Hws will usually open Friday mornings at7:00am. This first set is open a little earlier toget things going. Hws will close 12 days later onWednesday 1 minute before midnight. Thus therewill usually be two hws open from Friday throughWednesday. However, you should finish each hw be-fore the next hw opens. But if you get behind therewill be a 5 day grace period where you can catch up.

NotesIn Mathematics, and in ww (and also in most pro-

gramming languages), upper and lower case lettersare distinct. In most, if not all, ww problems in thisclass, all answers will be case sensitive! For example,use upper case letters when the answer is a Romannumber.

To maximize the benefit of the home work sets Irecommend that you ¡b¿answer the questions with-out using a calculator!¡/b¿ One of the purposes of thisclass is to increase your number sense and your men-tal computing ability. It may be hard at first to foregothe use of a calculator, but in the end it will pay manytimes in your career as an elementary school teacher.

One main purpose of this first WeBWorK set is tohelp you familiarize yourself with ww. Here are somehints on how to use ww effectively:

After first logging into ww change your password.Find out how to print a hard copy on the computer

system that you are going to use. Contact me if youhave any problems. Print a hard copy of this assign-ment.

Get to work on this set right away and answer thesequestions well before the deadline. That way youwon’t run out of time if you can’s solve a problem

right away, and it will give you a chance to to figureout what’s wrong if an answer is not accepted.

The primary purpose of the ww assignments in thisclass is to give you the opportunity to learn, for exam-ple by having instant feedback on your active solutionof relevant problems. Make the best of it!

The One Point ContestIf you find a mathematical error in one of these

home work problems, an exam, or a set of exam an-swers, and you bring it to my attention before I canfix it or announce a correction to the class, I will addone percentage to your final score in this class. I mayalso acknowledge your contribution to the class andmention your name. If you find a spelling or gram-matical error I will appreciate your letting me know,and you’ll earn my gratitude, but there are no extrapoints for finding non-mathematical errors.

Procrastination is hazardous!Peter Alfeld, JWB 127, 581-6842.¡IMG SRC=”http://www.math.utah.edu/ pa/images/image.gif”¿1.(10 pts) set1/4010s1p1.pg

This first question is just an exercise in entering an-swers into WeBWorK. It also gives you an opportu-nity to experiment with entering different arithmeticand algebraic expressions into WeBWorK and seeingwhat WeBWorK really thinks you are doing (as op-posed to what you believe it should think).Notice the buttons on this page and try them out be-fore moving to the next problem. Use the ”Back”Button on your browser to get back here whenneeded.

”Prob. List” gets you back to the list of all prob-lems in this set.

”Next” gets you to the next question in this set.”Submit Answer” submits your answer as you

might expect, but actually it does more. It shows youanswers and solutions (after the set closes), and it un-covers previously invisible hints. There is no limiton the number of times you can submit an answer,so you can also use it to see what WeBWorK thinksabout partial answers. It’s not effective simply to try alarge number of answers, but there is no harm in sub-mitting an answer even if you are not quite sure thatit’s correct, since if it is not you have an unlimited

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number of additional tries. On the other hand, it isusually more efficient to print your own private prob-lem set, work out the answers in a quiet environmentlike your home, and then sit down in front of a com-puter and enter your answers. If some are wrong youcan try to fix them right at the computer, or you maywant to go back and work on them quietly elsewherebefore returning to the computer.

Pressing on the ”Preview Answer” Button makesWeBWorK display what it thinks you entered in theanswer window. After using ”Preview” you can mod-ify your answer and use a ”Preview Again” button.This button is most useful, and grossly underused,when submitting complicated mathematical expres-sions.

”images” denotes the ordinary display mode onyour workstation. You can try the other modes, butfor our class ”images” works best. If you have usedWeBWorK before it probably had ”typeset” as its de-fault mode. In that case, things may look a little dif-ferent than you are used to.

”Logout” terminates this WeBWorK session foryou. You can of course log back in and continue.

”Feedback” enables you to send a message to me.If you use this way of sending e-mail I receive in-formation about your WeBWorK state, in addition toyour actual message.

The ”Help” Button transports you to an officialWeBWorK help page that has additional informationabout WeBWorK syntax.

”Problem Sets” transports you back to the pagewhere you can select a certain problem set. Whenyou do this particular problem in this first set, there isonly one set, but eventually there will be 13 of them.

For all problems in this course you will be ableto see the Answers to the problems after the duedate. Go to a problem, click on ”show correct an-swers”, and then click on ”submit answer”. You canalso download and print a hard copy with the an-swers showing. These answers are the precise stringsagainst which WeBWorK compares your answer. Ifthe answer is an algebraic expression your answerneeds to be equivalent to the WeBWorK answer, butit may be in a different form. For example if WeB-WorK thinks the answer is 2 ∗ a, it is OK for you totype a+a instead. If WeBWorK expects a numerical

answer then you can usually enter it as an arithmeticexpression (like 1/7 instead of .142857), and usuallyWeBWorK will expect your answer to be within onetenth of one percent of what it thinks the answer is.

Many of the problems (including this one) in thiscourse will also have solutions attached that you cansee after the due date by clicking on ”show solutions”followed by ”submit answers”. The solutions are texttyped by your instructor that gives more informationthan the ”answers”, and in particular often explainshow the answers can be obtained.

Now for the meat of this problem. Notice thatthe answer window is extra large so you can try outthings.

Type the number 3 here:.

Try entering other expressions and use the previewbutton to see what WeBWorK thinks you entered. Re-turn to this problem to try out things when you getstuck somewhere else.

Here are some good examples to try. Check themall out using the Preview button. (In later questionsyou will get to use what you learn here.) Never mindthat you may have already answered the correct an-swer 3. Once you get credit for an answer it won’t betaken away by trying other answers.

a/2b versus a/2/b versus a/(2b)a/b+c versus a/(b+c)a+b**2 versus (a+b)**2sqrt a+b versus sqrt(a+b)4/3 pi r**2 versus (4/3) pi r**2 (In other words, ifyou are not sure use parentheses freely.)

Note: WeBWorK will not usually let you enter al-gebraic expressions when the answer is a number, andit will only let you use certain variables when the an-swer is in fact an algebraic expression. So the abovewindow, and the opportunity for experimentation thatit offers is unique. Make good use of it!

Presumably this has been your first encounter withWeBWorK. Come back here to try things out andto refresh your memory if you get stuck somewheredown the line.

2.(10 pts) set1/4010s1p2.pgThe purpose of this exercise is to illustrate furtherthe use of the buttons on this page and to show youthe most common way in which WeBWorK processes

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partially correct problems. Try entering incorrect an-swers in the answer fields below, to see what hap-pens. (This time WeBWorK will reject algebraic ex-pressions since I told it to expect a numerical answer.)Type the number 4 here: .Type the number 5 here: .One of the trickier parts of ww is how to en-ter powers. For example, 23 = 2 × 2 × 2 =8. You can use the caret ””oradoubleasterisk” ∗∗”(inbothcaseswithoutthequotationmarks)toindicateexponentiation.Trythisbyentering8,2∗∗3,or23inthisbox : .

3.(10 pts) set1/4010s1p3.pgWeBWorK will usually consider a numerical answerto be correct if it is within one tenth of one percentof the answer that it has been given. However, youdo not usually have to enter a decimal approximation.WeBWorK will do simple arithmetic for you. The an-swer in this problem is 2

7 . You can enter this numberin various forms, e.g., 2/7, 4/14, 1/(7/2), 0.285714.Try it below. Also try to enter rough approximationslike 0.3, 0.29, 0.286, 0.2857, 0.28571 and see whichWeBWorK will accept.27 = .Occasionally, WeBWorK will insist on an answer be-ing a decimal expression. If so that expectation willbe stated in the problem. The next question is likethis. Try entering 2/7 and 0.28571, and see the differ-ence.27 = .

4.(10 pts) set1/4010s1p4.pgOK, we are now ready to start with some actual prob-lems. These address chapter 1 and the beginning ofchapter 2 of the textbook: problem solving, and num-ber systems.

You have one nickel, one dime, and one quar-ter. Using those coins you can form differentamounts of money. (Include the possibility of usingno coin, in which case your amount is zero cents.)

¡B¿Note:¡/b¿ This particular problem comes witha ¡b¿Hint¡/b¿ and a ¡B¿Solution¡/b¿. After you sub-mit your first answer, there will be a new button at thebottom of this page labeled ¡i¿Show Hint¡/i¿. Clickon it, then click on ¡I¿Submit Answer¡/i¿, and you’llsee the hint. Hints are of various levels of useful-ness. In this problem, it probably is not needed, in

other problems they may be essential. After the setcloses you can see a detailed solution. To see it goback to the problem, click on the new button ¡i¿ShowSolution¡/I¿, and click, once again, on ¡i¿Submit An-swer¡/I¿. You’ll see the Solution.

5.(10 pts) set1/4010s1p5.pg

You have one nickel, two dimes, and one quar-ter. Using those coins you can form differentamounts of money. (Include the possibility of usingno coin, in which case your amount is zero cents.)

6.(10 pts) set1/4010s1p6.pg

(This is a variation of problem B1 in section 1.1).Carol buys a certain number of items. She noticesthat the price of each item (in dollars) happens toequal the number of items she is buying. She spendsa total of 144 dollars (not counting tax). She bought

items.7.(10 pts) set1/4010s1p7.pg

(This is a variation of problem B3 in section 1.1).You open your textbook and you notice that the prod-uct of the page numbers happens to be 78680. Theleft page has number , and the right page haspage number .

8.(10 pts) set1/4010s1p8.pg

(This is a variation of problem B7 in section 1.1).The largest 8-digit number that you can form with thedigits 1, 1, 2, 2, 3, 3, 4, 4 is . The largest num-ber that you can form with the same digits, such thatthe 1’s are separated by at least one digit, the 2’s byat least two digits, the 3’s by at least 3 digits, and the4’s by at least 4 digits is . The smallest 8 digitnumber that satisfies the same constraints is .

9.(10 pts) set1/4010s1p9.pg

Questions that ask you to figure out the next num-ber in a sequence are used frequently on intelligenceand aptitude tests, and can be very simple or verycomplicated.

The next number in the sequence

1,2,3,4,5, . . .3

is .The next number in the sequence

1,3,5,7, . . .

is .The next number in the sequence

2,4,6,8, . . .

is .10.(10 pts) set1/4010s1p10.pg

The sequence of prime numbers is2,3,5,7,11, . . .

The next few prime numbers are, , , , , , , , .

11.(10 pts) set1/4010s1p11.pgThe square numbers are

1,4,9,16, . . .

The next few square numbers are, , , , , , , , , .

12.(10 pts) set1/4010s1p12.pgThe powers of 2 are

1,2,4,8, . . .

The next few powers of 2 are, , , , , , , , , .

13.(10 pts) set1/4010s1p13.pgThe ¡i¿triangular numbers¡/i¿ are

1,3,6,10,15, . . .

In our textbook they are defined graphically inproblem A6 of section 1.2.

The next few triangular numbers are: , , ,.

14.(10 pts) set1/4010s1p14.pg

You are playing the game of ”Adding On”. Oneplayer starts with a number from 1 to 12. Then theplayers take turns adding a number from 1 to 12. Theplayer who first says 100 wins. To be sure to win asthe first player you need to say the numbers

, , , , , , , .(Include the number 100 at the end.)15.(10 pts) set1/4010s1p15.pg

This is like the preceding problem, except that youneed to get all answers correct before getting credit.

You are playing the game of ”Adding On”. Oneplayer starts with a number from 1 to 8. Then theplayers take turns adding a number from 1 to 8. Theplayer who first says 80 wins. To be sure to win asthe first player you need to say the numbers

, , , , , , , , .(Include the number 80 at the end.)

16.(10 pts) set1/4010s1p16.pg

You are playing the game of ”Adding On”. Oneplayer starts with a number from 1 to 7. Then theplayers take turns adding a number from 1 to 7. Theplayer who must first say a number that’s greater thanor equal to 50 looses. To be sure to win as the firstplayer you need to say the numbers

, , , , , , .17.(10 pts) set1/4010s1p17.pg

You are playing the game of ”Adding On”. Oneplayer starts with a number from 1 to 9. Then theplayers take turns adding a number from 1 to 9. Theplayer who first says the number 70 wins. To be sureto win as the second player you need to say the num-bers

, , , , , , .18.(10 pts) set1/4010s1p18.pg

You buy a pot and its lid for a total of $ 11. The salesperson tells you that the pot by itself costs $ 10 morethan the lid. The price of the pot is $ andthe price of the lid is $ .

19.(10 pts) set1/4010s1p19.pg

This is problem B12 in section 1.1. of our text-book.

Find digits A, B, C, and D, that solve the followingcryptarithm.

ABCD×4 = DCBAThis works when ABCD = .20.(10 pts) set1/4010s1p20.pg

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This is a variation of Problem B5 in section 1.2 ofour textbook.

Let Pn be the n-th pentagonal number. Thus P1 = 1,P2 = 5, and P3 = 12. Moreover,

P4 = ,P5 = ,P6 = ,P9 = , andPn = .¡B¿Note¡/b¿ Your last answer in this problem will

be a polynomial expression in n. If you believe youhave the right answer but WeBWorK tells you it’swrong, use the preview answer button to see whatWeBWorK thinks you are saying. You may be miss-ing a suitable pair of parentheses.

21.(10 pts) set1/4010s1p21.pg

This is a variation of Problem B8 in section 1.2 ofour textbook.

The smallest number that can be expressed in twodifferent ways as the sum of two squares is .

22.(10 pts) set1/4010s1p22.pg

Consider the sequence1,11,21,1211,111221,312211, . . ..

The next number in this sequence is , andthe next one .

23.(10 pts) set1/4010s1p23.pg

The following few problems ask you to convertback and forth between the Roman and the Hindu-Arabic (or ¡i¿common¡/i¿) number system. Romannumbers are described in section 2.2 of the textbook.They are still used in some special applications, likeinscriptions on a building, numbering the front mat-ter in a book (including, for example, our textbook),or lettering on the face of a clock.

Enter the common equivalent for the following Ro-man numbers.

I = ,V = ,X = ,L = ,C = ,D = ,

M = .24.(10 pts) set1/4010s1p24.pg


VI = ,IV = ,LXIV = ,CCIV = ,MMVI = .25.(10 pts) set1/4010s1p25.pg

This is like the preceding problem, but you mustget all answers right before receiving credit.


XIV = ,LI = ,XLVI = ,CMLXXII = ,MCD = .26.(10 pts) set1/4010s1p26.pg

Enter the Roman equivalent of the following num-bers. Use capital letters, and leave no blanks. Don’tuse quotation marks. For example, simply enter thecapital letter I in the first box.

1 = ,2 = ,3 = ,4 = ,5 = .6 = ,7 = ,8 = ,9 = ,10 = .11 = .12 = .27.(10 pts) set1/4010s1p27.pg

Enter the Roman equivalent of the following num-bers. Use capital letters, and leave no blanks. Don’tuse quotation marks.

1 = ,5 = ,

5

10 = ,50 = ,100 = .500 = ,1000 = ,28.(10 pts) set1/4010s1p28.pg

Enter the Roman equivalent of the following num-bers. Use capital letters, and leave no blanks. Don’tuse quotation marks.

67 = ,1492 = ,2006 = ,49 = ,34 = .17 = ,999 = .29.(10 pts) set1/4010s1p29.pg

The next few problems deal with sets and set oper-ations.

Let S = {1,2,3,4} and T = {3,4,5,6,7}.Then S has elements and T has elements.

The set S ∪ T has elements, and S ∩ T haselements.

The set S−T has elements, and T −S haselements.

The largest element in S∪T is , and the largestelement in S∩T is .

The smallest element in S∪T is , and the small-est element in S∩T is .

30.(10 pts) set1/4010s1p30.pg

An international business has 22 employees. In ad-dition to everybody speaking Chinese, 8 people speakEnglish and 10 speak French. 3 speak both Frenchand English. speak neither French nor English.(However, among those they speak Spanish, German,Norwegian, Swahili, Latin, and Esperanto, in addi-tion to Chinese.)

31.(10 pts) set1/4010s1p31.pg

An international business has 18 employees. In ad-dition to everybody speaking Russian, 3 people speakonly English. 4 speak both English and French. 6speak neither French nor English. speak onlyFrench.

32.(10 pts) set1/4010s1p32.pg

This is problem B32 in section 2.1 of the text-book. A university professor asked his class of 42students when they had studied for his class the pre-vious weekend. Their responses were as follows:

9 had studied on Friday.18 had studied on Saturday.30 had studied on Sunday3 had studied on both Friday and Saturday10 had studied on both Saturday and Sunday6 had studied on both Friday and Sunday2 had studied on Friday, Saturday, and Sunday.

Assuming that all 42 students responded and an-swered honestly, you conclude that:

studied on Sunday, but not on either Friday orSaturday.

did all their studying in one day.did not study at all for this class last weekend.

33.(10 pts) set1/4010s1p33.pg

This is problem B33 in section 2.1 of the textbook.At an automotive repair shop, 50 cars were inspected.Suppose that 23 cars needed new brakes and 34 carsneeded new exhaust systems.

At least cars needed both new brakes and newexhaust systems.

At most cars needed both new brakes and newexhaust systems.No more than cars did not need any repairs.

34.(10 pts) set1/4010s1p34.pg

This is problem B34 in section 2.1 of the textbook.If 70% of all students take Science, 75% take SocialScience, 80% take mathematics, and 85% take Eng-lish, then at least % take all four subjects. At most

% take all four subjects.

Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c©UR6


Peter Alfeld.


due 6/7/06 at 11:59 PM.

Procrastination is hazardous!Peter Alfeld, JWB 127, 581-6842.The first fifteen problems in this home work set

concern sets. The next six questions are word prob-lems similar to some we’ve discussed in class. Thelast few problems concern the language of the deci-mal system, and conversions between different bases.

1.(10 pts) set2/4010s2p1.pgLet the two sets A and B be defined by

A = {1,2,3}, B = {3,4,5}.Furthermore, let

C = {1,2,3,4,5}D = {3}E = {1,2}F = {4,5}

In the following list, enter the letters C, D, E, F, toindicate the correct set.

A−B = .B−A = .B∩A = .B∪A = .2.(10 pts) set2/4010s2p2.pg

Let the two sets A and B be defined byA = {a,b,c,d}, B = {c,d,x,y}.

Furthermore, let

C = {a,b}D = {x,y}E = {c,d}F = {a,b,c,d,x,y}

As in the preceding problem, in the following list,enter the letters C, D, E, F, to indicate the correct set.

You need to use each letter. This time, you need toget everything correct before receiving credit.

B∪A = .B∩A = .A−B = .B−A = .3.(10 pts) set2/4010s2p3.pg

Let the two sets A and B be defined byA = {a,b,c,d}, B = {c,d,x,y}.

Furthermore, let

C = {a,b,x,y}D = {a,b,c,d}E = {a,b}F = /0

In the following list, enter the letters C, D, E, F, toindicate the correct set.

(A−B)∩ (B−A) = .(A−B)∪ (B−A) = .(A−B)∪ (A∩B) = .(A−B)− (A∩B) = .4.(10 pts) set2/4010s2p4.pg

Let S be a set. In the following list, enter the lettersS if the result is S, and the letter E if the result is theempty set.

S∪ /0 = .S∩ /0 = .S− /0 = ./0−S = .5.(10 pts) set2/4010s2p5.pg

Let S be a set. In the following list, enter the lettersS if the result is S, and the letter E if the result is theempty set. This time you need to get all answers cor-rect before receiving credit.

S∪S = .S∩S = .S−S = ./0− /0 = .6.(10 pts) set2/4010s2p6.pg

Let A and B be two sets. Enter the letter T if the pro-posed statement is true (for all sets A and B), and theletter F if it is false (for some sets A and B).

A∪B ⊆ A .A∩B ⊆ A .A−B ⊆ A .

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A−B ⊆ B .

7.(10 pts) set2/4010s2p7.pgLet A and B be two sets. Enter the letter T if the pro-posed statement is true (for ¡i¿all¡/i¿ sets A and B),and the letter F if it is false (for ¡i¿some¡/i¿ sets Aand B).

A∪B ⊆ A∩B .A∩B ⊆ A∪B .A∩B ⊆ A .A−B ⊆ A∪B .A−B ⊆ A∩B .

8.(10 pts) set2/4010s2p8.pgLet A, B, and C be three sets. Enter the letter T if theproposed statement is true (for ¡i¿all¡/i¿ sets A and B),and the letter F if it is false (for ¡i¿some¡/i¿ sets A andB).

A ⊆ B .B ⊆ A .A∪B ⊆ A∪B∪C .A∩B ⊆ A∩B∩C .A∩B∩C ⊆ A∩B .A∪B∪C ⊆ A∩B∩C .A∩B∩C ⊆ A∪B∪C .

9.(10 pts) set2/4010s2p9.pgIn the following questions, count all subsets, includ-ing the empty set, and the set itself. Thus, for exam-ple, the answer to the first question is 2.

The set S1 = {1} has subsets.The set S2 = {1,2} has subsets.The set S3 = {1,2,3} has subsets.The set Sn = {1,2,3, . . .n}, where n is any whole

number, has subsets.Let S = {1,2} and T the set of all subsets of S.

Then T has subsets.

10.(10 pts) set2/4010s2p10.pgThis is Problem B11 in section 2.1 of the textbook.Enter the letter T if the statement is true and the letterF if the statement is false.

The empty set is a subset of every set.The set {105,110,115,120, . . .} is an infinite set.

For all sets X and Y , either X ⊆ Y or Y ⊆ X .If A is an infinite set and B ⊆ A, then B also is an

infinite set.

For all finite sets A and B, if A∩B = /0, then thenumber of elements in A∪ B equals the number ofelements in A plus the number of elements in B.

11.(10 pts) set2/4010s2p11.pgIn the next few questions, enter the elements of aset in any order. You can separate the elements withblank space, but do not use commas! Remember thatas usual ww is case sensitive! This first problem isjust an exercise to enter sets as described.

For example, the correct sets in this problem areX = {b,c} and Y = {c}. You can enter X as ”b c”or ”c b” or ”cb” or ”bc” without the quotation marks,for example, but ¡b¿not¡/b¿ as ”BC” or ”b,c”. Apartfrom blank space, the only way to enter Y is as ”c”(without the quotation marks). Try it now!

The Cartesian product X ×Y of two sets X and Yis {(b,c),(c,c)}. Then X = { } and Y = { }.

12.(10 pts) set2/4010s2p12.pg

The Cartesian product X ×Y of two sets X and Y is{(a,b),(a,c),(b,b),(b,c),(c,b),(c,c)}. Then X = {

} and Y = { }.

13.(10 pts) set2/4010s2p13.pgThis is simply an exercise that shows you how to en-ter the empty set. Just leave the answer box blank.Try it now! Enter nothing and click on ”submit an-swer”. Some of the answers in the next question willbe empty sets, so just leave those boxes blank!

/0 = { },

14.(10 pts) set2/4010s2p14.pgThis is problem B13 of section 2.1 in the textbook.Remember to enter a blank string for the empty set.

Let R = {a,b,c}, S = {c,d,e, f}, T = {x,y,z}.Then

R∪S = { },R∩S = { },R∪T = { },R∩T = { },S∪T = { },S∩T = { },

15.(10 pts) set2/4010s2p15.pgSuppose X is a set with 4 elements and Y is a set with3 elements. Then X ×Y has elements.

2

Suppose X is a set with 5 elements and Y is a setwith 7 elements. Then X ×Y has elements.

Suppose X is a set with m elements and Y is a setwith n elements. Then X ×Y has elements.

16.(10 pts) set2/4010s2p16.pgThis is essentially problem B29 in section 2.1 of ourtextbook.

Your house can be painted in a choice of 11 exte-rior colors and 4 interior colors. Assuming that youchoose only one color for the exterior and one colorfor the interior, there are different ways of paint-ing your house.

17.(10 pts) set2/4010s2p17.pgAssuming a perfect grid system in Salt Lake City,there are shortest ways to walk from the cornerof 11th East and 11th South to the corner of 5th Eastand 6th South.

18.(10 pts) set2/4010s2p18.pgYou are playing the game of adding on. Players taketurns to add numbers from 1 to 64, and whoever firstsays 1100 wins. In order to be sure that you will winthe game you want to start with the number .

19.(10 pts) set2/4010s2p19.pgYou are playing the game of adding on. Players taketurns to add numbers from 1 to 66, and whoever firstsays 1609 wins. In order to be sure that you will winthe game you want to start with the number .

20.(10 pts) set2/4010s2p20.pgThe 1’s digit of 745 is .

21.(10 pts) set2/4010s2p21.pgThis is Problem B18 in section 2.2 of the textbook.The heights of five famous human-made structuresare related as follows:

The height of the Statue of Liberty is 65 feet morethat half the height of the Great Pyramid at Giza.

The height of the Eiffel Tower is 36 feet more thatthree times the height of Big Ben.

The Great Pyramid of Giza is 164 feet taller thanBig Ben.

The Leaning Tower of Pisa is 137 feet shorter thanBig Ben.

The total of all the heights is nearly half a mile. Infact, the sum of the five height is 2264 feet.

The height of Big Ben is feet, the height of theStatue of Liberty is feet, the height of the Eiffel

Tower is feet, The height of the Great Pyramid atGiza is feet, and the height of the Leaning Towerof Pisa is feet.

22.(10 pts) set2/4010s2p22.pgThe next few problems reinforce the language usedfor large numbers in the common system. Recall thefollowing names:

1 thousand = 1000 = 103

1 million = 1,000,000 = 106

1 billion = 1,000,000,000 = 109

1 trillion = 1,000,000,000,000 = 1012

1 quadrillion = 1,000,000,000,000,000 = 1015

1 quintillion = 1,000,000,000,000,000,000 = 1018

(The next few words in this sequence are 1 sex-tillion, 1 septillion, 1 octillion, and 1 nonillion. Wewon’t be using those in this class, and they arerarely introduced in elementary school classrooms.Of course, that should not stop you from telling themto your students if they are curious.)

According to the US National Debt Clock onMay 18, 2006, just before noon, the US public debtwas eight trillion, three hundred forty-one billion,nine hundred ninety-eight million, five hundred fifty-eight thousand one hundred seventy dollars. Enterthis number in decimal form, without spaces or com-mas, here: .

23.(10 pts) set2/4010s2p23.pgFor the following numbers, enter the number of zerosin the decimal representation of those numbers. Forexample, since one thousand is 1000, in the first itemyou enter the number 3.

one thousand:one trillion:one million:one billion:24.(10 pts) set2/4010s2p24.pg

For the following numbers, enter the number of zerosin the decimal representation of those numbers. Forexample, since one thousand is 1000, in the first itemyou enter the number 6.

one thousand thousands: ,one thousand millions: ,one million billions: ,one billion billion billions: ,

3

a thousand million billion trillion quadrillion quin-tillions .

25.(10 pts) set2/4010s2p25.pgMatch the numbers

A. one millionB. one billionC. one trillionD one quadrillionE one quintillionwith the numbers below. Thus, enter the letters A,

B, C, or D, E, as appropriate. You must get every an-swer right before receiving credit. For example, onethousand thousand thousands is a billion, so in thefirst item you want to enter the letter B.

one thousand thousand thousands: ,one thousand thousands: ,one million millions ,one thousand million billions ,one million billions ,

26.(10 pts) set2/4010s2p26.pg

In the next few exercises we will convert back andforth between base 10 and other base systems. Wewill use the notation they give in the textbook. Thus,for example,

123five = 1×52 +2×51 +3×50

= 25+10+3= 38= 38ten.

If the subscript is omitted it is understood to be 10,i.e., a missing subscript indicates a number expressedin our common decimal notation.

Convert the following numbers to their decimalequivalent. According to the above example, in thefirst box you need to enter the number 38.

123five = ,1234five = ,1111two = ,1111three = ,1111four = ,1111five = ,1111six = .27.(10 pts) set2/4010s2p27.pg

Express the number 100 in various other bases. Forexample, since100 = 64+32+4

= 26 +25 +0×24 +0×23 +22 +0×21 +0×20,

in the first box below you want to enter 1100100.100 = two,100 = three,100 = four,100 = five,100 = six,100 = seven,100 = eight,100 = nine,28.(10 pts) set2/4010s2p28.pg

Express the number 1000 in various other bases.1000 = two,1000 = three,1000 = four,1000 = five,1000 = six,1000 = seven,1000 = eight,1000 = nine,



Peter Alfeld.


due 6/14/06 at 11:59 PM.

Procrastination is hazardous!Peter Alfeld, JWB 127, 581-6842.This is a mixture of base conversion problems,

questions about the concepts of addition and multi-plication, and word problems.

1.(10 pts) set3/4010s3p1.pgThis is an exercise in base conversion, from base 6 tobase 10 to base 7. As in the textbook, we denote thebase by its English name as a subscript.

For example, since234six = 2×36+3×6+4

= 94= 94ten= 1×49+6×7+3= 163seven

in the first two boxes below you need to enter 94and 163.

234six = ten = seven.212six = ten = seven.310six = ten = seven.110six = ten = seven.

2.(10 pts) set3/4010s3p2.pgThis is an exercise in base conversion, from base 5 tobase 10 to base 8.

223five = ten = eight.231five = ten = eight.232five = ten = eight.


141five = ten = nine.310five = ten = nine.124five = ten = nine.


725eight = ten = nine.775eight = ten = nine.421eight = ten = nine.

5.(10 pts) set3/4010s3p5.pg

In the next few exercises you are asked to enter thebase that makes the equation true. For example, Prob-lem A19 of section 2.3 in the textbook asks: Whatbase makes the equation

32 = 44 ¡sub¿ ¡/sub¿ true?In this case, since

32 = 44seven,

you need to enter the string ”seven” (without the quo-tation marks). Try it now!

Here are some more exercises like this. What basesmake the following equations true?

100 = 244 ¡sub¿ ¡/sub¿112 = 422 ¡sub¿ ¡/sub¿6.(10 pts) set3/4010s3p6.pg

Here are some more exercises like this. What basesmake the following equations true?

24 = 220 ¡sub¿ ¡/sub¿196 = 400 ¡sub¿ ¡/sub¿7.(10 pts) set3/4010s3p7.pg

What bases make the following equations true?24 = 220 ¡sub¿ ¡/sub¿178 = 454 ¡sub¿ ¡/sub¿8.(10 pts) set3/4010s3p8.pg

What bases make the following equations true?301 = 455 ¡sub¿ ¡/sub¿166 = 325 ¡sub¿ ¡/sub¿9.(10 pts) set3/4010s3p9.pg

This is problem B16 from section 2.3. As we dis-cussed in class, the hexadecimal numeration system,used in computer programming, is a base sixteen sys-tem that uses the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,

1

A, B, C, D, E, and F. Change each of the followinghexadecimal numerals to base ten numerals:

213¡sub¿sixteen¡/sub¿ = ,A4¡sub¿sixteen¡/sub¿ = ,1C2B¡sub¿sixteen¡/sub¿ = ,420E¡sub¿sixteen¡/sub¿ = .10.(10 pts) set3/4010s3p10.pg

This is problem B17 from section 2.3. Write eachof the following base 10 numerals as base sixteen(hexadecimal) numerals.

375 = ¡sub¿sixteen¡/sub¿,2941 = ¡sub¿sixteen¡/sub¿,9520 = ¡sub¿sixteen¡/sub¿,24,274 = ¡sub¿sixteen¡/sub¿.11.(10 pts) set3/4010s3p11.pg

This exercise concerns some properties of addi-tion. For the following statements, enter the letter Cif the statement illustrates the commutative property,the letter O if it illustrates the closure of the wholenumbers under addition, the letter A if it illustratesthe associative property, and the letter Z if it illus-trates the identity property of zero. This question issomewhat like problem A4 of section 3.1.

5+0 = 5.11+47 = 47+11345+6789 is a whole number.7+(8+9) = (7+8)+9.

12.(10 pts) set3/4010s3p12.pg

For the following sets, enter the letter T (for true)if they are closed under addition, and the letter F (forfalse) if they are not. You must get all answers correctbefore receiving credit.

{3,4,5,6,7,8, . . .}the set of even numbersthe set of square numbersthe set of odd numbersthe set of prime numbersthe set of multiples of 4711{0}{0,1}13.(10 pts) set3/4010s3p13.pg

One of your best students asks you if empty set is

closed under addition. What do you tell her? Enter Yfor yes or N for no. .

14.(10 pts) set3/4010s3p14.pg

For the following sets, enter the letter T (for true)if they are closed under multiplication, and the letterF (for false) if they are not. You must get all answerscorrect before receiving credit.

{3,4,5,6,7,8, . . .}the set of even numbersthe set of square numbersthe set of odd numbersthe set of prime numbersthe set of multiples of 4711{0}{0,1}15.(10 pts) set3/4010s3p15.pg

This is problem B2 in set 3.1. For the followingpairs of sets, enter the letter T if it is true that

n(D)+n(E) = n(D∪E)

and the letter F otherwise.D = {a,c,e,g} and E = {b,d, f ,g}.D = {} and E = {}.D = {1,3,5,7} and E = {2,4,6}.

16.(10 pts) set3/4010s3p16.pg

This is problem B5 in set 3.1. Addition can be sim-plified by the associative property of addition. Forexample,

26+57 = 26+(4+53)= (26+4)+53= 30+53= 83.

Complete the following statements39+68 = 40+ = .25+56 = 30+ = .47+23 = 50+ = .17.(10 pts) set3/4010s3p17.pg

Suppose you are setting up an addition table inbase 6. For ease of exposition, let’s omit the subscript¡i¿six¡/i¿.

2

For example, in this problem, we write

5+3 = 12

which is short for

5six +3six = 12six.

Complete the following questions:2+2 = .2+3 = .2+4 = .2+5 = .3+3 = .3+4 = .3+5 = .4+4 = .4+5 = .5+5 = .18.(10 pts) set3/4010s3p18.pg

Suppose you are setting up a multiplication table inbase 6. For ease of exposition, let’s omit the subscript¡i¿six¡/i¿.

For example, in this problem, we write

5×3 = 23

which is short for

5six×3six = 23six.

Complete the following questions:2×2 = .2×3 = .2×4 = .2×5 = .3×3 = .3×4 = .3×5 = .4×4 = .4×5 = .5×5 = .19.(10 pts) set3/4010s3p19.pg

There’s a pond in your village. Somebody plants awater lily in it that doubles its size every day. Whileyou are off on a vacation the lily grows by a factor128. Your vacation was days long.

20.(10 pts) set3/4010s3p20.pgYour village depends for its food on a nearby fish-pond. One day the wind blows a water lily seed intothe pond, and the lily begins to grow. It doubles itssize every day, and its growth is such that it will coverthe entire pond (and smother the fish population) in45 days. You are away on vacation, and the villagersdepend on your vigilance and intelligence to handletheir affairs. Without you present they will take ac-tion about that lily only on regular work days, andonly when the lily covers half the pond or more. Thelily will cover half the pond days after it firststarts growing. The day it reaches that size happensto be a holiday....

21.(10 pts) set3/4010s3p21.pgThis is problem B21 in section 3.2. Complete thepattern. You can do this without using a calculator!WeBWorK does not understand commas as part of anumber, so enter your answers as numbers withoutblanks or commas, just like they are displayed here.

12345679×9 = 111111111.12345679×18 = 222222222.12345679×27 = .12345679×63 = .12345679×81 = .22.(10 pts) set3/4010s3p22.pg

I was unable to figure out why this problem B27 is insection 3.2 on multiplication and division. But it’s agreat question! Your students will love it!

Four men, one of whom committed a crime, saidthe following:

Bob: Charlie did itCharlie: Eric did itDave: I didn’t do itEric Charlie lied when he said I did it.If only one of the statements was true, then

was guilty. (Enter the name of the perpe-trator.)

Similarly, if only one of the statements was false,then was guilty.

23.(10 pts) set3/4010s3p23.pgThis is a variation of problem B24 in section 3.2.There are two four digit numbers that equal the cubeof the sum of their digits. The smaller is andthe larger is .

3

24.(10 pts) set3/4010s3p24.pgI’m looking for a two digit mystery number. It hasthe remainder 0 when divided by 2, the remainder 2when divided by 3, the remainder 0 when divided by5, and the remainder 1 when divided by 7. It is .

25.(10 pts) set3/4010s3p25.pgI’m looking for a two digit mystery number. It hasthe remainder 0 when divided by 2, the remainder 1when divided by 3, the remainder 3 when divided by5, and the remainder 4 when divided by 7. It is .

26.(10 pts) set3/4010s3p26.pgFor the following sets, enter the letter T if the set isclosed under multiplication, and the letter F if it isnot. Do you recognize a pattern?

The set of natural, or counting, numbers.The set of natural numbers with 11 removed.The set of natural numbers with 21 removed.The set of natural numbers with 31 removed.The set of natural numbers with 41 removed.The set of natural numbers with 51 removed.27.(10 pts) set3/4010s3p27.pg

This is an exercise in converting from one base toanother. Of course, you can always accomplish thisby converting from the first base to base 10, and thenconverting from base 10 to the second base. How-ever, in this case, it is easy to convert without usingbase 10 at all.

0four = two1four = two2four = two3four = two2110four = two1200four = two2233four = two28.(10 pts) set3/4010s3p28.pg

This is like the last problem, except you go theother way.

1101001two = four10010110two = four10101011two = four29.(10 pts) set3/4010s3p29.pg

Here is a puzzle that will intrigue your students. Youhave a glass of water and a glass of ink. The glasses

have the same size, and you have as much water asyou have ink. You move a spoonful of ink from theink to the water, and stir it well. Then you take aspoonful of the ink and water mixture and transfer itto the ink. After this process, do you have more ink inthe water glass, more water in the ink glass, or do youhave as much water in the ink as you have ink in thewater? Once you figure out the answer, can you thinkof a convincing argument why it is what it is? Enterhere the word ”ink”, ”water”, or ”both”, (with-out the quotation marks) corresponding respectivelyto the three possibilities mentioned above.

By the way, you can do a related activity whereinstead of the two glasses of fluid you have a pileof pennies and another pile with an equal number ofdimes. Think of the pennies and dimes as water andink molecules (although, of course, ink usually is asolution of some dye in water.)

30.(10 pts) set3/4010s3p30.pg

You students will be fascinated by this problem.You can modify it in various ways to suit your gradelevel.

In a well known story the inventor of the gameof chess was asked by his well pleased King whatreward he desired. ”Oh, not much, your majesty”,the inventor responded, ”just place a grain of rice onthe first square of the board, 2 on the next, 4 on thenext, and so on, twice as many on each square as onthe preceding one. I will give this rice to the poor.”(For the uninitiated, a chess board has 64 squares.)The king thought this a modest request indeed andordered the rice to be delivered.Let fn denote the number of rice grains placed onthe first n squares of the board. So clearly, f1 = 1,f2 = 1 + 2 = 3, f3 = 1 + 2 + 4 = 7, and so on. How¡b¿does¡/b¿ it go on? Compute the next two values offn: f4 = f5 =Ponder the structure of this summation and then enteran algebraic expression that definesfn = .Supposing that there are 25,000 grains of rice in apound, 2000 pounds in a ton, and 6 billion peopleon earth, the inventor’s reward would work out to ap-proximately tons of rice for every per-son on the planet. Clearly, all the rice in the kingdom

4

would not be enough to begin to fill that request. The story has a sad ending: feeling duped, the king causedthe inventor of chess to be beheaded.



Peter Alfeld.


due 6/21/06 at 11:59 PM.

Procrastination is hazardous!This home work set is on terminology, powers, and

word problems.Peter Alfeld, JWB 127, 581-6842.1.(10 pts) set4/4010s4p1.pg

The first few exercises in this set deal with terminol-ogy. Enter the appropriate words without spaces orquotation marks. For example, in the first answerbox, enter the word ”sum”, and in the second, enterthe word ”addition”, in both cases without the quota-tion marks.

In the equation c = a+b, c is the of a and b,and the process of computing c is called . Wesay that we are a and b.

Similarly, in the equation c = a× b, c is theof a and b, and the process of computing c is called

. We say that we are a and b.2.(10 pts) set4/4010s4p2.pg

In the equation c = a− b, c is the of a and b,and the process of computing c is called . Wesay that we are b from a.

Similarly, in the equation c = a÷ b, c is theof a and b, and the process of computing c is called

. We say that we are a by b. In this con-text, a is called the and b is called the .

3.(10 pts) set4/4010s4p3.pg

The expression am is called a . a is theand m is the .

4.(10 pts) set4/4010s4p4.pg

This is similar to the preceding problems. The an-swer in the first box is the word ”plus”.

The expression a+b is pronounced a b.

The expression a−b is pronounced a b.The expression a×b is pronounced a b.The expression a÷b is pronounced a by b.5.(10 pts) set4/4010s4p5.pg

In this question you are asked to enter a mathe-matical expression defined by the given phrase. Forexample, in the first answer box you need to enter theexpression ”a+b” (without the quotation marks).

The sum of a and b is .The quotient of a and b is .The product of a and b is .The difference of a and b is .6.(10 pts) set4/4010s4p6.pg

This is similar to the previous question.a to the power b is .7.(10 pts) set4/4010s4p7.pg

This is similar to the previous question.The quotient of the sum of a and b and the differ-

ence of a and b is.

8.(10 pts) set4/4010s4p8.pg

This is similar to the previous question.The product of the sum and the difference of a and

b is .9.(10 pts) set4/4010s4p9.pg

This is similar to the previous question.The product of a whole number x and the next

whole number is .10.(10 pts) set4/4010s4p10.pg

This is similar to the previous question.The difference of a whole number x and its square

is .11.(10 pts) set4/4010s4p11.pg

The next few problems deal with powers and rules ofpowers.

Suppose n is any natural number. Then0n = .1n = .n0 = .

1

n1 = .12.(10 pts) set4/4010s4p12.pg

Suppose a, n and m are whole numbers. Thenanam = ab where b = .(an)m = ab where b = .13.(10 pts) set4/4010s4p13.pg

Let a, b, m, n be natural numbers. Enter T for thefollowing statements if they are true for all choices ofthe variables, and F if they are false for some choicesof the variables.

ab = ba .ab = ab .aman = am+n .aman = amn .a0 = 0 .0a = 0 .(am)n = am+n .(am)n = amn .(a+b)2 = a2 +b2 .(a+b)(a−b) = a2 −b2 .14.(10 pts) set4/4010s4p14.pg

Find two natural numbers a = and b =such that

1 < a < band

ab = ba.

15.(10 pts) set4/4010s4p15.pgAs we discussed in class, I recommend that youmemorize certain powers that occur frequently.These include the squares of the numbers from 1 to20, the cubes of the numbers from 1 to 10, and thepowers for 2 from 20 to 210. You should do theseproblems without a calculator.

The following few problems reinforce familiaritywith these and some other powers. We start with thesquares of 11 through 20. (Squares of single digitnumbers are of course part of the multiplication ta-ble.)

112 = .122 = .132 = .142 = .

152 = .162 = .172 = .182 = .192 = .202 = .16.(10 pts) set4/4010s4p16.pg

Next we have the cubes of 1 through 10. (Squaresof single digit numbers are of course part of the mul-tiplication table.)

13 = .23 = .33 = .43 = .53 = .63 = .73 = .83 = .93 = .103 = .17.(10 pts) set4/4010s4p17.pg

and the powers of 2.21 = .22 = .23 = .24 = .25 = .26 = .27 = .28 = .29 = .210 = .18.(10 pts) set4/4010s4p18.pg

OK, let’s see if you can remember these powers theother way. For example, since 8 = 23, in the first twoanswer boxes below you will want to enter 2 and 3.For all of these questions you want an exponent thatis greater than 1, i.e., b > 1. In all cases, a and b arewhole numbers and b > 1.

Answer the following questions.8 = ab where a = and b = .343 = ab where a = and b = .361 = ab where a = and b = .

2

19.(10 pts) set4/4010s4p19.pgThis problem is similar to the preceding one. As be-fore, you want b > 1.

125 = ab where a = and b = .128 = ab where a = and b = .196 = ab where a = and b = .20.(10 pts) set4/4010s4p20.pg

You buy a new hard drive for your computer. It has acapacity of 200 billion bytes. The manufacturer ad-vertises this disk as a ¡i¿200 gigaByte drive¡/i¿. Ascomputers do, your computer considers a gigaByteto mean 230 bytes. So it tells you that your new drivehas a capacity of gigaBytes. (Compute the an-swer and round to the nearest natural number.) Youconsider switching to a different vendor.

21.(10 pts) set4/4010s4p21.pgHere are some more specific powers. Ideally, youshould build enough number sense that you can tellthe answers with little calculation, but if you get stuckyou can compute the prime factorization of the givennumber and go from there.

81 = ab = cd where a = , b = , c =d = , and b > d > 1.

243 = ab where a = and b = .22.(10 pts) set4/4010s4p22.pg

As before, you want all exponents to be greater than1.

256 = ab = cd = e f

where a = , b = , c = , d = ,e = , f = , and b > d > f > 1.

23.(10 pts) set4/4010s4p23.pgMore of the same:

512 = ab = cd

where a = , b = , c = , d = ,and b > d > 1.

24.(10 pts) set4/4010s4p24.pgThis also works for powers of 10.

1,000,000 = ab = cd = e f

where a = , b = , c = , d = ,e = , f = , and b > d > f > 1.

25.(10 pts) set4/4010s4p25.pgThis is a modified version of Problem B12 in section3.3. Ask yourself how the sum 1+2+3 compares tothe sum 13 + 23 + 33. Try several more such exam-ples, and conjecture a relations ship between the sum1+2+3+ . . .+n and the sum 13 +23 +33 + . . .+n3.Finally, compute the sum

13 +23 +33 + . . .+103 = without evaluatingany of the cubes.

26.(10 pts) set4/4010s4p26.pgThis is a modified version of Problem B14 in section3.3. Order these numbers

322, 414, 910, 810

from smallest to largest using properties of powersand mental methods, and enter them by increasingsize. You can enter them by typing, for example,3**22, you do not need to compute decimal versionsof these numbers.

27.(10 pts) set4/4010s4p27.pgThe last few problems in this set are simple wordproblems that you can pose to your students. They areadapted from the AA problem solving card deck thatI passed around in class. As you know, these prob-lems solving cards where one of the two key items inmy own elementary school teaching, the other beinga supply of base 10 blocks.

The sun is about 93,120,000 miles form the earth.The moon is about 240,000 miles from the earth. Thesun is times as far from the earth as the moon.

28.(10 pts) set4/4010s4p28.pgSome Chinese typewriters have 5400 characters. TheEnglish typewriter has 44 characters. There arefewer characters on an English typewriter than on aChinese typewriter.

29.(10 pts) set4/4010s4p29.pgAlice is younger than Bob. The difference in theirages is 3 years, the sum is 25 years. Alice isand Bob is years old.

3

30.(10 pts) set4/4010s4p30.pgpairs of whole numbers have 75 as their prod-

uct. Consider two pairs the same if one can be ob-tained from the other by switching the numbers, forexample, the pair (5,15) is the same as the pair (15,5).

31.(10 pts) set4/4010s4p31.pgBob worked twice as long as Jan. Jan worked 1 hour

more than Kim. Kim worked 2 hours less than Pedro.Pedro worked three hours. Bob worked hours.

32.(10 pts) set4/4010s4p32.pgI’m thinking of a two digit number that equals twicethe product of its digits. It is .

The number of problems in this home work setequals ab where a = , b = , and b > 1.



Peter Alfeld.


due 6/28/06 at 11:59 PM.

Procrastination is hazardous!The first twelve problems of this set are exercises

in mental computation. You can easily solve theseproblems with a calculator, but try to get the answerjust mentally, using nothing but your mind, and noaids like calculators or paper and pencil. The chal-lenge is on, if my cobwebbed brain can handle theseproblems, then so can yours! There are quite a fewaddition problems, and you may be getting tired ofthem, but being skilled at ad mental addition is mostuseful in daily living, so take those problems as achance to practice.

Problems 13–25 concern written algorithms for thefour basic operations.

Problems 26–35 reinforce your familiarity with theterminology of the four operations.

Problems 36–40 are simple word problems.Peter Alfeld, JWB 127, 581-6842.1.(10 pts) set5/4010s5p1.pg

This is problem 11 of set 4.1 in our textbook. Five ofthe following six numbers were rounded to the near-est thousand, then added to produce an estimated sumof 87,000.

5228 14286 7782 19628 9168 39228The number was not included in this estimate.

2.(10 pts) set5/4010s5p2.pgThis is similar to the preceding problem. Five of thefollowing six numbers were rounded to the nearestthousand, then added to produce an estimated sum of126,000.

24998 4088 49299 30139 14261 18272The number was not included in this estimate.

3.(10 pts) set5/4010s5p3.pgThis is problem 17 in section 4.1. Guess what num-bers a and b can be used to make the following equa-tions true:

a4 = 6561, a = ,b5 = 16,807, b = .

4.(10 pts) set5/4010s5p4.pgThis is similar to the preceding problem.

a4 = 2401, a = ,b6 = 262144, b = .

5.(10 pts) set5/4010s5p5.pgOf the following additions one is correct, the othersare incorrect. Enter the reference letter of the additionthat is correct. For example, in the additions below,the first one can’t be right because the answer is muchtoo large, and the second can’t be right since the sumof two odd numbers is even. The third one can’t beright because the last digit of the sum must be 0. Sothe correct one must be D. Enter the letter D below.

A. 347+453 = 10,000,000,B. 347+453 = 797,C. 347+453 = 802,D. 347+453 = 800.

is correct.

6.(10 pts) set5/4010s5p6.pgOf the following additions, one is correct, the othersare incorrect. Enter the reference letter of the addi-tion that is correct.

A. 587+524 = 1112,B. 587+524 = 1111,C. 587+524 = 1110,D. 587+524 = 2223.

is correct.

7.(10 pts) set5/4010s5p7.pgOf the following additions, one is correct, the othersare incorrect. Enter the reference letter of the addi-tion that is correct.

A. 4533+5829 = 20725.B. 4533+5829 = 10361,C. 4533+5829 = 10362,D. 4533+5829 = 10363,

is correct.

8.(10 pts) set5/4010s5p8.pgIn this and the next three problems, mentally compute

1

the indicated sums, and try to get them all without us-ing any aids like a calculator or paper and pencil.

17+18 = ,37+75 = ,278+396 = ,1459+2609 = .

9.(10 pts) set5/4010s5p9.pgMentally compute the following sums.

19+9 = ,29+28 = ,58+85 = ,144+256 = .


9+19 = ,29+58 = ,88+175 = ,264+526 = .


13+19 = ,33+58 = ,92+175 = ,268+526 = ,795+1579 = .

12.(10 pts) set5/4010s5p12.pgNow try your mind on these two digit products.Again, avoid the use of any aids. (This particularproblem may be a little challenging.)

18×29 = ,34×44 = ,51×63 = ,77×81 = .

13.(10 pts) set5/4010s5p13.pgThis is a simple review of the standard algorithm foraddition. Fill in the variables as required. All an-swers are single digit whole numbers. If you wouldordinarily leave a location blank enter the digit 0.The columns in this problem are set apart from eachother for increased clarity, but of course we are sim-ply adding the two three digit numbers 647 and 568.

In the additiona b6 4 7

+ 5 6 8− − − −c d e f

,

a = ,b = ,c = ,d = ,e = ,f = .Note: In this and the following problems, you

don’t need to fill in these boxes in the given sequence.You could just do the addition and fill in the boxes asyou figure out their contents. But it might be easierto work it out on paper and then simply transfer thedigits in the given sequence.

14.(10 pts) set5/4010s5p14.pgHere is another addition problem. It’s similar to theprevious problem, but you have more terms in thesum, and you need to get everything right before get-ting credit.


+ 6 4 7+ 2 4 5+ 3 8 8− − − −c d e f

,

a = ,b = ,c = ,d = ,e = ,f = .

15.(10 pts) set5/4010s5p15.pgIn this exercise we consider two different subtractionalgorithms. Both were discussed in class. We firstillustrate them with an example. To compute

814−277 = 537

the ¡i¿standard¡/i¿ algorithm proceeds as follows:2

107 6 0 146 8 6 1 6 4

− 2 7 7− − −5 3 7

The missing addend based Washington Algorithmis a little more subtle conceptually but cleaner:

8 1 4− 2 7 7

1− − −5 3 7

Now try both algorithms on the subtraction526−287 = 239.

The standard algorithm becomes

ab 6 c d6 5 6 2 6 6

− 2 8 7− − −2 3 9

wherea = ,b = ,c = ,d = ,and the Washington algorithm becomes

5 2 6− 2 8 7

e f− − −2 3 9

wheree = andf = .Note: The standard algorithm becomes quite

messy for this problem.16.(10 pts) set5/4010s5p16.pg

Use the Washington algorithm for the subtraction403−198 = 205.

4 0 3− 1 9 8

e f− − −2 0 5

wheree = andf = .

17.(10 pts) set5/4010s5p17.pgThe multiplication and division algorithms are stillimportant, despite the wide availability of calcula-tors, because they apply with simple modificationsto the multiplication and division of polynomials, asyou learned in Math 1010 and 1050. There are dif-ferent ways to arrange the notation of multiplication,we use then one given in the textbook, and illustratedby this example:

3 4× 9 6− − −2 0 4

3 0 6− − − −4 2 6 4

Here, 6× 34 = 204 and 9× 34 = 306. The 306 isshifted one column to the left, because they are 306tens. Alternatively, you can think of it as 3060 =90× 34, with the zero omitted. The required addi-tion can be carried out by they standard addition al-gorithm. It’s particularly simple in this case, sinceevery column contains a zero.

Note that we multiply the first factor with the digitsof the second factor. We could also reverse the rolesof the first and second factor, and should certainly doso if it is convenient. However, in the example be-low also multiply the first factor with the digits of thesecond.

Fill in the digits in the following calculation:3

2 7× 3 8− − −a b c

d e f− − − −g h i j

We geta = ,b = ,c = ,d = ,e = ,f = ,g = ,h = ,u = , andj = .18.(10 pts) set5/4010s5p18.pg

This is similar to the preceding problem, but this timeyou need to get everything right before getting credit.

Fill in the digits in the following calculation:

5 6× 6 4− − −a b c

d e f− − − −g h i j

We geta = ,b = ,c = ,d = ,e = ,f = ,g = ,h = ,u = , andj = .19.(10 pts) set5/4010s5p19.pg

Of course, insight can be gained by carrying out ouralgorithms in a base other than 10. In this additionproblem, everything is base 6.

Find the digits in the following addition

a b3 4 5

+ 1 4 2− − −c d e

sixHere,a = ,b = ,c = ,d = ,e = ,20.(10 pts) set5/4010s5p20.pg

Now do this subtraction in base six, using the Wash-ington algorithm.

Find the digits in the following subtraction

3 2 1− 1 4 5

a b− − −c d e

sixHere,a = ,b = ,c = ,d = ,e = ,21.(10 pts) set5/4010s5p21.pg

Next, a multiplication in base six.Find the digits in the following multiplication

3 4× 2 3− − −a b c

d e f− − − −g h i j

sixHere,a = ,b = ,c = ,d = ,e = ,

4

f = ,g = ,h = ,i = ,j = ,22.(10 pts) set5/4010s5p22.pg

This is similar to the preceding problem, but you needto get all digits correct before getting credit.

Find the digits in the following multiplication

4 5× 3 2− − −a b c

d e f− − − −g h i j

sixHere,a = ,b = ,c = ,d = ,e = ,f = ,g = ,h = ,i = ,j = ,23.(10 pts) set5/4010s5p23.pg

The long division algorithm is the most complicatedprocedure taught in elementary school. There is aschool of thought that it ought to be omitted since it’spainful to carry out, and it can be replaced with theaid of a calculator. However, in algebra, which stu-dents will study soon after leaving elementary school,the long division of polynomials is an important pro-cedure. It will be hard to understand if the studentshave not had some exposure to long division of num-bers before thinking about the same for polynomials.The two subjects are closely connected because whilenumbers are sums of powers of the variable, we ex-press a number as a sum of powers of 10. We won’tdiscuss long division of polynomials in this class, butsince that will be on the horizon for your studentsyou do want to discuss with them the long division ofwhole numbers.

Here is an example showing the division

782÷23 = 34

that illustrates the notation we discussed in class.

3 4− − −

2 3 | 7 8 26 9

− −9 29 2

− − −0

Find the digits in the following division

a b− − −

3 7 | 8 5 1c d− − −e f gh i j− − −

0

Here,a = ,b = ,c = ,d = ,e = ,f = ,g = ,h = ,i = ,j = ,

24.(10 pts) set5/4010s5p24.pgOf course, in a division there may be a non-zero re-mainder. The remainder will show up at the bottomof the calculation where previously we had a zero.Fill in the digits in the following long division withremainder:

5

a b− − −

3 7 | 9 9 0c d− − −e f gh i j− − −

k lHere,a = ,b = ,c = ,d = ,e = ,f = ,g = ,h = ,i = ,j = ,k = ,l = ,25.(10 pts) set5/4010s5p25.pg

I know you are itching to do a long division in a baseother than 10, but I’m sorry to disappoint you! In-stead, as the grand finale of these algorithm prob-lems, here is another division with remainder whereyou need to get every digit right before getting credit.

a b− − −

3 6 | 8 7 7c d− − −e f gh i j− − −

k lHere,a = ,b = ,c = ,d = ,e = ,f = ,

g = ,h = ,i = ,j = ,k = ,l = ,26.(10 pts) set5/4010s5p26.pg

Here are some more questions on terminology. Inall cases, the variables a, b, and c stand for arbitrarywhole numbers.

The rule a + b = b + a is the property of.

The rule (a+b)+c = a+(b+c) is the prop-erty of .

The rule a × b = b × a is the property of.

The rule (a×b)×c = a×(b×c) is the prop-erty of .

27.(10 pts) set5/4010s5p27.pgThe rule a(b + c) = ab + ac is the law of mul-tiplication over addition.

28.(10 pts) set5/4010s5p28.pgIn the expression a÷ b, a is the and b is the

.29.(10 pts) set5/4010s5p29.pg

The difference of 12 and 3 is .The sum of 12 and 3 is .The quotient of 12 and 3 is .The product of 12 and 3 is .

30.(10 pts) set5/4010s5p30.pgIn this problem, enter the appropriate word corre-sponding to one of the four basic arithmetic opera-tions.

4 is the of 8 and 2.10 is the of 8 and 2.6 is the of 8 and 2.16 is the of 8 and 2.31.(10 pts) set5/4010s5p31.pg

Adding the sum of 3 and 4 to the product of 4 and 5gives .

Subtracting the difference of 12 and 8 from the dif-ference of 100 and 50 gives .

Adding the quotient of 36 and 9 to the product of36 and 9 gives .

6

Multiplying the quotient of 7 and 2 with the quo-tient of 2 and 7 gives .

32.(10 pts) set5/4010s5p32.pgAdding the difference of 14 and 9 to the product of 7and 8 gives .

Adding the quotient of 126 and 9 to 8 gives .The quotient of 126 and 14 is .The sum of the product of 7 and 8 and the differ-

ence of 14 and 9 is .9 to the power 7 equals . (You may need to

use a calculator for this last problem.)

33.(10 pts) set5/4010s5p33.pgI have a mystery number. I form the difference of thatnumber and 9. Then I form the product of the resultand 4. I add the result to 6. Then I form the differenceof the result and 9. I get 21. My mystery number is

.

34.(10 pts) set5/4010s5p34.pgI have a mystery number. I form the sum of that num-ber and 2. The I multiply the result with the productof 4 and 6. I form the difference of the result and 18.Finally, I form the quotient of the result and 2, and Iget 75. My mystery number is number is .

35.(10 pts) set5/4010s5p35.pgI have a mystery number. I form the difference of thatnumber and 4. Then I form the product of the resultand 7. I add the result to 9. Then I form the differenceof the result and 4. I get 33. My mystery number is

.

36.(10 pts) set5/4010s5p36.pgFinally, a few word problems.

The sum of the ages of three children is 22 years.Susan is the oldest. Mark is not the youngest. Bobby

is six years younger than the oldest who is ten yearsold. Mark is years old.

37.(10 pts) set5/4010s5p37.pgThe fewest number of coins you would need so thatyou are able to pay a charge of any amount less than25 cents with exact change is .

38.(10 pts) set5/4010s5p38.pgCal earned $ 14. He gave $ 4 to Ray and half of whatwas left to Alice. He kept $ .

39.(10 pts) set5/4010s5p39.pgThe following two problems are examples of ”sim-ple Hindu Algebra”, quoted on page 528 of ”TheStory of Civilization”, v.1., by Will Durant, Simonand Schuster, 1935. The problems are approximately1,800 years old. (Mental pursuits by women werediscouraged at that time. You ponder the significanceof ficticiously directing the questions at women. Du-rant does not comment on this issue.)Out of a swarm of bees one fifth part settled on aKadamba blossom; one third on a Silihindra flower;three times the difference of those numbers flew tothe bloom of a Kutaja. One bee, which remained,hovered about in the air. Tell me, charming woman,the number of bees .

40.(10 pts) set5/4010s5p40.pgHere is the other problem from ”The Story of Civi-lization”:Eight rubies, ten emeralds, and a hundred pearls,which are in thy ear-ring, my beloved, were pur-chased by me for thee at an equal amount; and thesum of the prices of the three sort of gems was threeless than half a hundred; tell me the price of each,auspicious woman.Enter the price of one pearl , the price of oneemerald , and the price of one ruby .



Peter Alfeld.


due 7/5/06 at 11:59 PM.

Procrastination is hazardous!Most problems in this set concern prime numbers,

factors, divisibility rules, greatest common factorsand least common multiples. There are also a fewword problems, and some Nim Puzzles.

Peter Alfeld, JWB 127, 581-6842.1.(10 pts) set6/4010s6p1.pg

The first few problems in this set deal with primenumbers. Everybody should know the prime num-bers below 100. So to begin with, list them in in-creasing sequence in the following boxes. Of course,you can look them up in any table of prime numbers,but for the sake of the exercise, obtain them startingwith 2 and computing them mentally.

1st2nd3rd4th5th6th7th8th9th10th11th12th13th14th15th16th17th18th19th20th21st

22nd23rd24th25th

2.(10 pts) set6/4010s6p8.pgNow continue your quest and compute mentally theprime numbers between 100 and 200. There are 21of them, they are the 26-th through the 46-th primes.

26th27th28th29th30th31st32nd33rd34th35th36th37th38th39th40th41st42nd43rd44th45th46st

3.(10 pts) set6/4010s6p9.pgSome numbers look like primes at first sight, butaren’t. For the following composite numbers, en-ter the prime factors in increasing sequence. Notethat prime may be repeated. For example, since49 = 7× 7, in the first two boxes below you’ll wantto enter 7 and 7.

49 = × .91 = × .121 = × .143 = × .1001 = × × .

4.(10 pts) set6/4010s6p2.pgLook back at the table of primes you created in thelast question. There are prime numbers that areless than 100.

1

A prime twin is a pair of prime numbers that differby 2. For example, 3 and 5 form a prime twin, andso do 5 and 7. There are a total of prime twinsformed by prime numbers less than 100.

5.(10 pts) set6/4010s6p3.pgAs we discussed in class, it is unknown whether theGoldbach Conjecture that every even number greaterthan 2 can be written as the sum of two prime num-bers is true or false.

In this and the following problem you are asked towrite several even numbers as sums of pairs of primenumbers. In each case, if the two prime numbers aredistinct, enter the smaller first. For example, since8 = 3 + 5, and this is the only way to write 8 as thesum of two prime numbers, in the first two answerboxes below you need to enter 3 and 5, in that se-quence.

Write the following numbers as sums of two primenumbers:

8 = + .12 = + .Some even numbers can be written in more than

one way as a sum of two prime numbers. For exam-ple,

10 = 3+7 = 5+5.

When entering multiple pairs like this, start with onethat has the smallest first term. For example, in thenext four answer boxes you need to enter 3, 7, 5, 5,in that sequence:

10 = + = + .Similarly,14 = + = + .6.(10 pts) set6/4010s6p4.pg

The smallest number n that can be written in threedifferent ways as a sum of two prime numbers is

n = = + = + = + .(Remember to enter the smaller prime number

first, and sort pairs by increasing first prime number.)7.(10 pts) set6/4010s6p5.pg

The smallest number n that can be written in four dif-ferent ways as a sum of two prime numbers is

n = = + = + = + =+ .

(Remember to enter the smaller prime numberfirst, and sort pairs by increasing first prime number.)

8.(10 pts) set6/4010s6p6.pgRecall that a number is called abundant if it is smallerthan the sum of its proper factors, perfect if it equalsthat sum, and deficient if is greater than that sum.

For the following numbers enter the letter A if theyare abundant, P if the are perfect, and D if they aredeficient.

2: .3: .4: .6: .12: .20: .28: .36: .100: .9.(10 pts) set6/4010s6p7.pg

The smallest perfect number is .The next smallest perfect number is .The next smallest perfect number is .

10.(10 pts) set6/4010s6p10.pgSome smallish numbers have lots of divisors. Below,enter the divisors of 720 ordered by increasing size,starting with 1, and ending with 720. As usual, togain the maximum benefit from this exercise, do itmentally. Note that the divisors come in pairs. Forexample, since 2 is a divisor, 720/2 = 360 is also adivisor. Thus you can fill in the blanks below work-ing inwards from both ends, or you can make a listand transfer it to the boxes below.

, , , , , , , , , , ,, , , , , , , , , , ,, , , , , , , .

11.(10 pts) set6/4010s6p11.pgYou might think that 720 has more divisors than anyother three digit number. I did until I checked, but Iwas wrong. 720 has 30 divisors. There is one (andonly one) three digit number with 32 divisors. It is

.12.(10 pts) set6/4010s6p12.pg

The next few problems concern divisibility ideas. Forreference, here are some of the rules we discussed inclass. A number n (expressed in the base 10 system)is divisible by:

2: if the last digit of n is 0, 2, 4, 6, or 8.2

3: if the sum of the digits of n is divisible by 3.4: if the number formed by the last two digits of n

is divisible by 4.5: if the last digit of n is 0 or 5.6: if it’s divisible by 2 and 3, according to the

above rules.7: if we can carry out the division algorithm with-

out remainder. There is no simpler rule I know of, butof course that division isn’t hard.

8: if the number formed by the last three digits ofn is divisible by 8.

9: if the sum of the digits of n is divisible by 9.10: if the last digit of n is 0.11: if 11 divides the difference of the sum of the

digits of n whose place values are odd powers of 10and the sum of the digits whose place values are evenpowers of 10.

any composite number: if n is divisible by all pow-ers of prime factors that occur in the prime factoriza-tion.

Also recall that m|n means that m divides n.For the following statements, decide whether the

following statements are true using the above divis-ibility rules. You shouldn’t need to use a calculatoror do a manual division. Enter the letter T if they aretrue, and the letter F if they are false.

9 | 12345678911 | 12345678913.(10 pts) set6/4010s6p13.pg

This problem is similar to the preceding one, but youneed to get everything right before receiving credit.

2 | 12533 .5 | 9040 .10 | 16680 .9 | 1278 .3 | 2450 .11 | 18927 .11 | 12793 .14.(10 pts) set6/4010s6p14.pg

This problem is similar to the preceding one.6 | 37082 .6 | 25566 .6 | 17499 .12 | 1332 .24 | 5760 .18 | 39744 .

22 | 40062 .7 | 14161 .15.(10 pts) set6/4010s6p15.pg

Try to answer this question using the divisibility rulesto rule out smaller numbers.

The smallest prime number greater than 1000 is.

The smallest prime number greater than 10000 is.

16.(10 pts) set6/4010s6p16.pgThis problem is similar to the preceding one.

1021 is prime. The next prime is .17.(10 pts) set6/4010s6p17.pg

This problem is similar to the preceding one.12011 is prime. The next prime is .18.(10 pts) set6/4010s6p18.pg

The next few problems involve greatest common fac-tors and least common multiples.

The greatest common factor of 12 and 18 isand their least common multiple is .

19.(10 pts) set6/4010s6p19.pgThe greatest common factor of 21 and 35 is andtheir least common multiple is .

20.(10 pts) set6/4010s6p20.pgFind the greatest common factors of the pairs of num-bers below. For the more complicated examples, Irecommend you use the Euclidean Algorithm and acalculator. The point of these exercises is to illustratethe the Euclidean algorithm always works, and thatit gives the answer easily when a prime factorizationwould be hard to obtain.

gcf(1073,2183) =gcf(3431,1457) =gcf(4757,4891) =

21.(10 pts) set6/4010s6p21.pgThis is similar to the preceding problem.gcf(43531,78011) =gcf(69541,152143) =gcf(46003,94319) =

22.(10 pts) set6/4010s6p22.pgOf course one can compute the least common multi-ple of more than two numbers. It’s useful to be awareof the least common multiples of the first few naturalnumbers.

3

lcm(2,3) = .lcm(2,3,4) = .lcm(2,3,4,5) = .lcm(2,3,4,5,6) = .lcm(2,3,4,5,6,7) = .lcm(2,3,4,5,6,7,8) = .lcm(2,3,4,5,6,7,8,9) = .lcm(2,3,4,5,6,7,8,9,10) = .

23.(10 pts) set6/4010s6p23.pgThis is a modification of problem B19 in section 5.1.of the textbook. The smallest natural number n forwhich n2 −n+41 is not prime is n = .

24.(10 pts) set6/4010s6p24.pgThis is problem B20 of section 5.1. I’m a two digitnumber less than 40. I’m divisible by only one primenumber. The sum of my digits is prime, and the dif-ference between my digits is another prime. I am

. My big sister has the same properties. Sheis . I also have a little brother with those prop-erties. He is .

25.(10 pts) set6/4010s6p25.pgThis is problem B20 of section 5.2 in the textbook.Tilda’s car gets 34 miles per gallon and Naomi’s gets8 miles per gallon. When traveling from Washing-ton, D.C., to Philadelphia, they both used a wholenumber of gallons of gasoline. Philadelphia ismiles from Washington.

26.(10 pts) set6/4010s6p26.pgThis is problem B24 of section 5.2 in the textbook.The theory of biorhythm states that there are three”cycles” to your life:

The physical cycle: 23 days longThe emotional cycle: 28 days longThe intellectual cycle: 33 days longIf your cycles are together today, they will be to-

gether again days from now. It will be a while,you’d better enjoy this day while it lasts!

27.(10 pts) set6/4010s6p27.pgThis is problem B28 of section 5.2 in the textbook.See also the story on page 227.

Ramanujan observed that 1729 is the smallestnumber that can expressed as the sum of two cubesin two ways.

Find natural numbers a, b, c, and d such that1729 = a3 +b3 = c3 +d3

where the numbers are ordered such thata < b, and a < c < d.

a = .b = .c = .d = .28.(10 pts) set6/4010s6p28.pg

And more miscellaneous word problems... You can’tdo enough of those when you teach your class. Mostof the following problems are taken from the AAProblem Solving Card Deck.

Twenty-two people went to the theater. Three peo-ple rode to the theater in each car, and seven peopletook the bus. On the way home, four people rode ineach car, and took the bus.

29.(10 pts) set6/4010s6p29.pgA 60 foot board is cut into two pieces. On piece is 12feet longer than the other piece. The shorter piece is

feet long, and the longer is feet long.30.(10 pts) set6/4010s6p30.pg

You are going to drive 480 miles today at an aver-age speed of 60 miles per hour. Thus you are go-ing to drive for a total of hours. The diameter ofthe wheels on your car (including tires) is 25 inches.Thus each wheel is going to turn a total of timestoday.

31.(10 pts) set6/4010s6p31.pgOne ounce of Rochos perfume sells for $ 25. Notcounting any volume discounts, the price per gallonof the perfume is $ .

32.(10 pts) set6/4010s6p32.pgPaul is 12. His father is 33. years ago the fatherwas four times as old as the son.

33.(10 pts) set6/4010s6p33.pgMaureen spend $ 25.60 for a shirt and a tie. The shirtcost $ 12 more than the tie. The tie cost $ .

34.(10 pts) set6/4010s6p34.pgFinally here are a few Nim puzzles. Nim is a greatgame to play and analyze in elementary school. Itillustrates the use of symmetry and the utility of thebinary system. Kids are motivated by the competitive

4

aspect, and they love to play against and beat theirparents, friends, and relatives. In the process theynaturally do quite a bit of mental arithmetic.

For reference, here are the rules. You placematches (or other suitable objects) in five rows, con-taining 1, 2, 3, 4, and 5 matches respectively, as il-lustrated below. Other initial configurations are pos-sible. Two players keep taking turns each removingas many matches as he or she likes, but only fromone row at each time. Whoever takes the last matchwinds.

As we discussed in class, you always want to playsuch that after your move there is an even number of1s, 2s, and 4s. Your opponent will have to destroythat attribute, you rebuild it, and eventually you willget to play such that there are zero 1s, 2s, and 4s, andyou win.

In order not to give away the answer in these prob-lems I will refer to the plural ”matches” even whenyou need to remove a single match.

Suppose you start with the standard starting con-figuration:

1. |2. | |3. | | |4. | | | |5. | | | | |

You have three possible winning moves. Enterthem by increasing row numbers. They are:

Remove matches from row , orRemove matches from row , orRemove matches from row .35.(10 pts) set6/4010s6p35.pg

Suppose that after each of you removed one matchform the third row you face this configuration:

1. |2. | |3. |4. | | | |5. | | | | |

This time you have just one possible winningmove.

You remove matches from row , or

36.(10 pts) set6/4010s6p36.pgThe winning move in this configuration

1. |2. | |3. |4. | |5. | | | | |

is to remove matches from row .

37.(10 pts) set6/4010s6p37.pgOf course, the principles we’ve discussed don’t workjust for the standard initial configuration. There arethree winning moves in this configuration:

1. |2. | |3. | | |4. | | | |5. | | | | |6. | | | | | |

They are:Remove matches from row , orRemove matches from row , orRemove matches from row .

38.(10 pts) set6/4010s6p38.pgSuppose you are playing against a savvy opponent,and you are facing the loosing configuration:

1. |2. | |3. | |4. | | | |5. | | | | |

In hopes of creating the maximum amount of con-fusion for your opponent you decide to play such thatthere is an odd number of 1s, 2s, and 4s. To that end

you remove matches from row , oryou remove matches from row .

39.(10 pts) set6/4010s6p39.pgThe principles we’ve discussed also work for largerpowers of 2. Indeed, kids get intrigued if you letthem set up the initial configuration giving them awide range of numbers of rows or matches. Supposeyour student set up this configuration:

5

1. | | | | | |2. | | | |3. | | | | | | | | |4. | | | | | | |5. | | | | | | | | | | |

You have three winning moves:Remove matches from row , orRemove matches from row , orRemove matches from row .40.(10 pts) set6/4010s6p40.pg

And now for the grand finale. Suppose your brighteststudent sets up this particular winning configurationand challenges you to start and play against her:

1. | | | | | | | | | | |2. | | | | | | | |3. | | | | | | |4. | | | |5. | | |6. | |7. |

To make it a worthy challenge you decide to playsuch that after your play there is an odd number ofeach power of 2, i.e., of 1s, 2s, 4s, and 8s. You havetwo possible moves. They are:

Remove matches from row , orRemove matches from row .



Peter Alfeld.


due 7/12/06 at 11:59 PM.

Procrastination is hazardous!This set is focused on fractions. It also contains a

large number of word problems.Peter Alfeld, JWB 127, 581-6842.1.(10 pts) set7/4010s7p1.pg

The first few problems are exercises in fraction arith-metic. Your particular fractions are chosen by ran-dom, and it is possible that an answer is a whole num-ber, in which case you would enter 1 as the denom-inator. There is also a tiny chance that you get thesame problem more than once.

The expression129 +

512

can be written in lowest terms as a fractionab

wherea= and b=

2.(10 pts) set7/4010s7p2.pgThe expression

103

+5

12can be written in lowest terms as a fraction

ab

wherea= and b=


65 − 5


ab

wherea= and b=


63 − 5


ab

wherea= and b=


211 × 11


ab

wherea= and b=


129× 3

10can be written lowest terms as the fraction

ab

wherea= and b=


87 ÷ 11

6can be written in lowest terms as the fraction

ab

wherea= and b=


23 ÷ 11

12can be written lowest terms as the fraction

ab

where1

a= and b=9.(10 pts) set7/4010s7p9.pg

The expression107 − 5

649 + 12


ab

wherea=

and b= .10.(10 pts) set7/4010s7p10.pg

The expression109 − 3

1287 + 2


ab

wherea=

and b= .11.(10 pts) set7/4010s7p11.pg

The expression 12 × 2

3 × 34 × 4

5 × 56 × 6

7 can be writtenin lowest terms as the fraction a

b wherea= and b=

12.(10 pts) set7/4010s7p12.pgLet x

y be a fraction (reduced to lowest terms) that sat-isfies

12 +

xy =

56 .

Then x= and y=13.(10 pts) set7/4010s7p13.pg

Let xy be a fraction (reduced to lowest terms) that sat-

isfies37× x

y =65.

Then x= and y=14.(10 pts) set7/4010s7p14.pg

The expression 12 ÷ 2

3 ÷ 34 ÷ 4

5 ÷ 56 ÷ 6

7 can be writtenin lowest terms as the fraction a

b wherea= and b=

15.(10 pts) set7/4010s7p15.pgYour new mountain bike with the fancy Shimano

equipment has two front cogs with 53 and 42 teeth,respectively, and six rear cogs with 12, 13, 15, 17, 20,23, and 26 teeth, respectively. Thus you have a totalof 2× 7 = 14 gears available. Enter below the gearssorted by increasing ratio of the numbers of front andrear teeth. This corresponds to increasing distancecovered by your bike for each rotation of the pedals.For example, the lowest gear is obtained when youchoose the largest wheel (26 teeth) in the rear, andthe smallest (42 teeth) in front. In that case, the rearwheel will make 42/26 rotations for every rotation ofthe pedals. Thus the first answers you should enterbelow are 26 and 42.

Gear 1: Rear: , Front: .Gear 2: Rear: , Front: .Gear 3: Rear: , Front: .Gear 4: Rear: , Front: .Gear 5: Rear: , Front: .Gear 6: Rear: , Front: .Gear 7: Rear: , Front: .Gear 8: Rear: , Front: .Gear 9: Rear: , Front: .Gear 10: Rear: , Front: .Gear 11: Rear: , Front: .Gear 12: Rear: , Front: .Gear 13: Rear: , Front: .Gear 14: Rear: , Front: .

16.(10 pts) set7/4010s7p16.pgSuppose the wheels on your fancy new bike (de-scribed in the previous problem) have a diameter of29 inches, including the tires. Then in lowest gear,every time you turn the pedals once, your bike willmove a distance of inches. In highest gear, yourbike will move inches.

17.(10 pts) set7/4010s7p17.pgYou take your fancy new bike of the last two prob-lems to the salt flats and you ride 13 miles in yourhighest gear, paddling constantly. You pedals turn

times, and your wheels turn times duringthe trip. (Round your answers to the nearest wholenumbers.)

18.(10 pts) set7/4010s7p18.pgThe next few problems are exercises in fraction termi-nology. Insert the correct words. When in doubt, use”fraction” instead of ”rational number”, and ”equal”

2

instead of ”equivalent” (all without the quotationmarks).

In the 34 , the number 3 is the and the

number 4 is the .19.(10 pts) set7/4010s7p19.pg

Of the two fractions 57 and 9

7 , the first is a frac-tion and the second is an fraction.

20.(10 pts) set7/4010s7p20.pgThe two fractions 6

9 and 1015 are .

21.(10 pts) set7/4010s7p21.pgIn simplest , the fraction 6

9 is to 23 .

22.(10 pts) set7/4010s7p22.pgIn lowest , the fraction 6

9 is to 23 .

23.(10 pts) set7/4010s7p23.pgConsider the two fractions 5

6 and 29 . The denominator

of the first fraction is , the denominator of thesecond is , and the least common denominatoris .

24.(10 pts) set7/4010s7p24.pgThe least common denominator of the fractions

12 ,

13 ,

14 ,

15 ,

16

is and the sum of these fractions, in lowestterms, is a

b where a = and b = .

25.(10 pts) set7/4010s7p25.pgInsert the appropriate words in these sentences:

A common denominator of two fractions is anycommon of the two denominators, and the leastcommon denominator is the least common ofthe two denominators.

26.(10 pts) set7/4010s7p26.pgThe general formula for the addition of two fractionsis

ab +

cd =

xy

where x = and y = .The general formula for the subtraction of two

fractions isab − c

d =xy

where x = and y = .

27.(10 pts) set7/4010s7p27.pgThe general formula for the multiplication of twofractions is a

b × cd =

xy

where x = and y = .The general formula for the division of two frac-

tions is ab ÷ c

d =xy

where x = and y = .28.(10 pts) set7/4010s7p28.pg

It takes you 8 hours to mow a certain lawn. (It’s a biglawn!) It takes your brother 7 hours to mow that samelawn. Working together it takes you and your brothera/b hours to mow that lawn, where a = andb = . (Express your answer in simplest form.)

29.(10 pts) set7/4010s7p29.pgIt takes you 4 hours to paint a certain room. Workingwith your sister, it takes the two of you 20/9 hours topaint that same room. By herself, your sister wouldtake hours to paint that room.

30.(10 pts) set7/4010s7p30.pgI have the following data from Herman Melville’sbook Moby Dick. (Once you get into it, it’s a spell-binding read!) The crew of a whaling ship where paida certain percentage of the proceeds of the voyage,called a ”lay”. This was expressed as a unitary frac-tion (with a numerator of 1). Thus the 300th lay was1/300 of the proceeds from the trip. For example,a typical captain’s lay was the seventeenth lay. Af-ter a long back and forth, Ishmael, the hero of thestory, is hired as a junior crew member for the 300thlay. Later, his friend, Queequeg, an accomplishedharponeer, is hired for the ninetieth lay. Queequegmakes times as much money as Ishmael.

31.(10 pts) set7/4010s7p31.pgOn a certain fishing boat, the captain is paid the thirdlay and her mate the fifth lay. Ordinary crew mem-bers are paid the tenth lay. Assuming that the ownerof the boat (who is not a crew member) makes somepositive profit, the largest possible number of regularcrew members is .

Note: I made up those numbers, but got the ideafor this problem from the book ”The Hungry Ocean:A Swordboat Captain’s Journey” by Linda Greenlaw.

3

It has more realistic and very detailed information onthis topic.

32.(10 pts) set7/4010s7p32.pgYou’ll want to use your calculator for this problem.

π is the ratio of the circumference and the diame-ter of a circle. It’s an irrational number that cannot beexpressed exactly as a decimal. However, an approx-imate value is

π ≈ 3.14159265358979.

A well known, and pretty good, rational approxima-tion of π is 22/7. In fact, the error (i.e., the absolutevalue of the difference of π and its approximation)in this approximation is . A somewhat betterapproximation is 223/71. The error in that approxi-mation is .

Note: ww expects your answer to be within onetenth of one percent of the true answer. To be safespecify at least 4 digits beyond the leading zeros.

33.(10 pts) set7/4010s7p33.pgThis is problem B16 in section 6.2 of our textbook.Grandma is planning to make a red, white, and bluequilt. One third is to be red and two fifths are to bewhite. Since the total area of the quilt is going to be30 square feet, an area of square feet is goingto be blue.

34.(10 pts) set7/4010s7p34.pgThis is problem B17 in section 6.2 of our textbook.A recipe for cookies will prepare enough for threeseventh of Ms. Jordan’s class of 28 students. If shemakes three batches she can feed extra stu-dents.

35.(10 pts) set7/4010s7p35.pgThis is problem B18 in section 6.2 of our textbook.Karl wants to fertilize his 6 acres. Since it takes 8 2

3bags of fertilizer for each acre he will need to buy

bags.36.(10 pts) set7/4010s7p36.pg

This is a variation of problem B19 in section 6.2 of

our textbook. During one evening Kathleen devoted2/5 of her study time to mathematics, 3/20 of herstudy time to Spanish, 1/3 of her time to biology,and the remaining 35 minutes to English. It was along evening! Indeed, Kathleen studied for a total of

minutes, of which she spent minutes onmathematics.

37.(10 pts) set7/4010s7p37.pgThis is an elaboration of problem B20 of section 6.3in our textbook. According to the Container Recy-cling Institute, 57 billion aluminum cans were recy-cled in the United States in 1999. That amount wasabout 5/11 of the total number of aluminum cans soldin the United States in 1999. Thus aluminumcans were sold in the United States in 1999.

Assuming that the US population in 1999 was 270million people this works out to an average ofcans per person in 1999. (Round that last number tothe nearest integer).

38.(10 pts) set7/4010s7p38.pgThis is problem B24 of section 6.3 in our textbook.You place one full container of flour on one panof a balance scale, and a similar container 3/4 fulland a 1/3 pound weight on the other pan. The pansbalance. You conclude that your container of flourweighs pounds.

39.(10 pts) set7/4010s7p39.pgThis is problem B26 of section 6.3 in our textbook.An airline passenger fell asleep halfway to her desti-nation. When she awoke, the distance remaining washalf the distance traveled while she slept. She wasasleep during of the entire trip.

40.(10 pts) set7/4010s7p40.pgThis is problem B34 of section 6.3 in our textbook.At a dinner with guests, every two guestsshared a bowl of rice, every three guests shared abowl of broth, every four guests shared a bowl offowl, and 65 bowls were used altogether.



Peter Alfeld.


due 7/19/06 at 11:59 PM.

Procrastination is hazardous!This set has some more fraction problems, but is

focused on decimals, ratios, proportions, and percentproblems.

Even if not specifically so stated, please enter allfractions in lowest terms.

Some of the questions ask you to round an answerto the closes whole number. That means you replacethem by the smaller whole number if the first digit be-yond the decimal point is 0, 1, 2, 3, o4 4, and by thenext larger whole number otherwise. For example,you’d round 7.43 to 4 and 7.501 to 8.


We continue our quest for a thorough understandingof fraction arithmetic. In the following problems, en-ter the numerator and denominator of the the answerin lowest terms.

As a warm up consider the problem12 − 1

3 = / .The result of the subtraction is 1

6 , so you enter thenumbers 1 and 6.

2.(10 pts) set8/4010s8p2.pgLet

z =1213

.

Then z = / .The result of the subtraction is 3

2 , so you enter thenumbers 3 and 2.

3.(10 pts) set8/4010s8p3.pgLet

z =132357

.

Then z = / .4.(10 pts) set8/4010s8p4.pg

Let

z =

132357

Then z = / .5.(10 pts) set8/4010s8p5.pg

A pretty good way to approximate the square root of 2is as follows. Start with a reasonable approximation,call it s1. Use it to compute a better approximation

s2 =s2

1 +22s1

.

Then use the same procedure to a yet better approxi-mation

s3 =s2

2 +22s2

.

Continue doing this. Thus you obtain a sequences1, s2, s3, . . .

of terms that get closer and closer to the square rootof 2. The sequence goes on forever, but of course wehave to stop somewhere, and actually, for this partic-ular sequence, sooner rather than later.

Suppose we start withs1 = 1.

Thens2 =

12 +22

=32.

and

s3 =

(32)2

+22× 3

2=

1712 .

Next we get get

s4 =

(1712

)2+2

2× 1712

.

Compute and enter s4 = / .Use your calculator to compute the error, s4 −

√2,

and enter it here. You will need to enter it as adecimal number with at least four digits beyond theleading zeros.

Notes: s5 will have ten leading zeros after the dec-imal point. Pretty impressive! The above is called

1

an iteration. Your calculator uses a similar iterationto compute square roots. This particular iterationis called Newton’s Method applied to the equationx2 −2 = 0.

6.(10 pts) set8/4010s8p6.pgThis is similar to the preceding problem, except thatthis time we approximate the square root of 3. Thecorresponding iteration is

s2 =s2

1 +32s1

, s3 =s2

2 +32s2

, etc.

Suppose we start again withs1 = 1.

Thens2 = / ,s3 = / ,s4 = / , ands5 = / .Email me your guess of how one might approxi-

mate the square root of a number p.7.(10 pts) set8/4010s8p7.pg

We discussed in class that mixed numbers are thebane of humankind. Try your hand on this example.Let

z =32

351

3.

Then z = / ,8.(10 pts) set8/4010s8p8.pg

Here is another example. Let

z =33

561

7.

Then z = / ,9.(10 pts) set8/4010s8p9.pg

You are working for an airline where everybody’ssalary is cut by one third. Your salary before the cutis $ 36,000. After the cut it is $ . Your coworkermentions that after the cut his salary is $ 26,000. Be-fore the cut it was $ .

10.(10 pts) set8/4010s8p10.pgThis is Problem B31 in section 6.3 of our textbook.The sum of two numbers is 18, and their product is40. Without finding the numbers (which are not sim-ple whole numbers anyway), compute the sum of the

reciprocals of the two numbers which is , and thesum of the squares of the two numbers which is .

11.(10 pts) set8/4010s8p11.pgThe next few problems explore the relationships be-tween fractions and decimal expressions. A fractioncan be written as a repeating decimal (where a termi-nating decimal is a special case of a repeating deci-mal), and a repeating decimal can be converted to afraction. A terminating decimal can be written as afraction with a power of 10 in the denominator. Afraction can be written as a terminating decimal if inlowest terms its denominator contains only 2 and 5as prime factors. The basic idea of fractions is sim-ply to carry the hierarchy of powers of 10 1s to tents,hundredths, thousandths, and so on.

In these problems, if the the answer is a fractionwhere you specify numerator and denominator sepa-rately, please enter the fraction in lowest terms.

Write 25 as a decimal: .

Enter the decimal 0.34 as a fraction in simplestform: / .

12.(10 pts) set8/4010s8p12.pgWrite 3

32 as a decimal: .Enter the decimal 0.185 as a fraction: / .13.(10 pts) set8/4010s8p13.pg

Write the repeating decimal z = 0.71 as a fraction.z = / .Write the repeating decimal z = 0.1234 as a frac-

tion.z = / .14.(10 pts) set8/4010s8p14.pg

Write the repeating decimal z = 0.185 as a fraction.z = / .Write the repeating decimal z = 2063.428571 as a

fraction.z = / .15.(10 pts) set8/4010s8p15.pg

The fraction z = 17 can be written as the repeating dec-

imalz = 0.abcde f

where the digits a, b, c, d, e, and f area = ,b = ,c = ,

2

d = ,e = ,f = .16.(10 pts) set8/4010s8p16.pg

The fraction z = 19 can be written as the repeating dec-

imalz = 0.a

where the digit a isa = .17.(10 pts) set8/4010s8p17.pg

The fraction z = 111 can be written as the repeating

decimalz = 0.ab

where the digits a and b area = andb = ,18.(10 pts) set8/4010s8p18.pg

For the following fractions, indicate with the letter Tif their decimal equivalents terminate, and with theletter R, if they repeat, but do not terminate.

135 : ,37 : ,1716 : ,148 : ,2135 : ,1111024 : .

19.(10 pts) set8/4010s8p19.pgThis question is similar to the preceding one, exceptthat this time you need to get all answer correctly be-fore receiving credit.

For the following fractions, indicate with the letterT if their decimal equivalents terminate, and with theletter R, if they repeat, but do not terminate.

1315 : ,1243250 : ,4416 : ,3723 : ,1211100 : ,1

211 : .

20.(10 pts) set8/4010s8p20.pgEnter the following numbers as decimals:

three tenths ,four hundred eighty nine millionths ,

one one hundred thousandth ,two billionths .21.(10 pts) set8/4010s8p21.pg

Enter the following numbers as decimals:thirteen tenths ,four hundred eighty nine thousandths ,a million thousandths ,six billion billionths .22.(10 pts) set8/4010s8p22.pg

Environmental Protection Agency guidelines holdthat mercury rates of higher than 3.5 parts per billionin a person’s body pose a possible health threat. Sup-posing that a person weighing 135 pounds has thatcritical level of mercury, the total amount of mercuryin that person’s body is pounds.

23.(10 pts) set8/4010s8p23.pgConsider the fraction 1

7 represented as a decimal withinfinitely many (or at least more than 1265) digits be-yond the decimal point. The 1265 -th of those digitsis . (Of course you don’t to have to compute orwrite out that decimal expression to 1265 digits.)

24.(10 pts) set8/4010s8p24.pgThis is a modified version of problem B22 in section7.2. If you look closely you’ll notice that the state-ment there can’t possibly be right. Why not?

In any case, observing that9

13 = 0.692307

you conclude that the 999th digit to the right of thedecimal point is . The 1154-th digit is .

25.(10 pts) set8/4010s8p25.pgThis is a modification of Problem A1 in section 7.3.You can enter your answers as fractions like 1/3.

You have 10 boys and 19 girls in your class.The ratio of boys to girls is .The ratio of girls to boys is .The fraction of students who are boys in your class

is .The fraction of students who are girls in your class

is .26.(10 pts) set8/4010s8p26.pg

You worked 5.5 hours and were paid $ 59.4. Yourhourly wage was $ . (Enter your answers as anumber, not an arithmetic expression.)

3

27.(10 pts) set8/4010s8p27.pgLet x be defined by the proportion

x25.5 =

1624 .

Then x = . (Enter your answers as a number,not an arithmetic expression.)

28.(10 pts) set8/4010s8p28.pgThis is problem B12 from section 7.3. Grape juiceconcentrate is mixed with water in a ratio of 1 partconcentrate to 3 parts water.Thus you can makeounces of grape juice from a 10-ounce can of concen-trate. (Enter your answers as a number, not an arith-metic expression.)

29.(10 pts) set8/4010s8p29.pgThis is a variation of problem B21 from section 7.3.Your school has 1020 students. The average class sizeis 34 students. Your principal has been lobbying toreduce the class size to 30 students. The cost of run-ning one class is $ 35400. The cost per student underthe old system is $ , and under the proposed newsystem it would be $ .

Enter your answers as numbers, rounded to thenearest integer.

30.(10 pts) set8/4010s8p30.pgThis is a variation of Problem A23 of section 308.Astronomers believe that the universe is about 14 bil-lion years old. To put this long period of time intoperspective it is often compared to an ordinary 24hour day. So assume the universe was formed exactly24 hours ago. WeBWorK expects your answers to becorrect to at least three digits, so to be safe computethem to four digits, including digits beyond the deci-mal point. You probably want to use your calculatorfor these calculations.

One hour in the 24 hour day corresponds toyears of real time, one minute corresponds toyears, and one second corresponds to years.

The earth was formed 5 billion yeas ago. In the 24hour day this corresponds to a time hours ago.

The first human like beings appeared 2.6 millionyears ago. In our model day that was secondsago.

Let’s say civilization started 10,000 years ago.That corresponds to a time seconds ago.

31.(10 pts) set8/4010s8p31.pgThis is problem B19 of section 7.3. You have a mapdrawn to a scale of 1/8 inch representing 70 feet. Onthat map the shortest way from your house to the gro-cery store measures 23 7

16 inches. Thus the distancefrom your house to the grocery store is miles.

32.(10 pts) set8/4010s8p32.pgYou are hiking along the California coast and wonderabout the height of a particular Giant Redwood tree.You are 5 feet and nine inches tall and your shadow is12 feet long. The shadow of the tree is 432 feet long.The tree is feet tall.

33.(10 pts) set8/4010s8p33.pgYour European friend tells you proudly that he hiked34 kilometers that day. You remember that a kilome-ter is one hundred thousand centimeters, and an inchis 2.54 centimeters. So you figure that your friendwalked a distance of miles that day.

34.(10 pts) set8/4010s8p34.pgThe next few problems on this set deal with simplepercentage problems.You make $ 95000 a year, and you get a 10% raise.Your new salary is $ . (Salaries in theseproblems are randomly chosen and range from $20,000 to $ 100,000.)The next year you get a 10% cut. Your new salaryis $ . That is % less than your initialsalary.

35.(10 pts) set8/4010s8p35.pgAfter a 5% raise your new salary is $ 84000. Beforethe raise your salary was $ .

36.(10 pts) set8/4010s8p36.pgThis is a true story! The Math Department recentlyobtained a new building. We had to evaluate furniturebids from various manufacturers that offered variousdiscounts. One manufacturer offered a ”30+30” per-cent discount. This means they cut 30 percent offthe list price and then took another 30 percent off thediscounted price. Other manufacturers offered var-ious single discounts. All other things being equalyou prefer the single discount if it is greater thanpercent.

37.(10 pts) set8/4010s8p37.pgThis is Problem B36 from section 7.4. Think of a

4

whole number. Add 35. Multiply by 10. Find 20% of your last result. Find 50 % of the last number.Subtract the number you started with. You get .Can you explain what’s happening?

38.(10 pts) set8/4010s8p38.pgThis is Problem B40 from section 7.4. One fourthof the world’s population is Chinese, and one fifth ofthe rest is Indian. % of the world population areIndian.

39.(10 pts) set8/4010s8p39.pgYou are responsible for a reservoir holding rainbowtrout and you want to estimate how many trout arein the reservoir. To that end you catch 82 fish, markthem, and release them. A week later you catch 115fish. Of those, 16 turn out to be marked. Assumingthat each time you caught the fish they were selectedentirely by random, and no fish died or were addedto the reservoir, (assumptions which are not realistic)you figure that the population of your fish is approxi-mately . (Round to the nearest whole number.)

40.(10 pts) set8/4010s8p40.pgThis is a modification of discussion problem 10 onpage 330 of the textbook. Suppose your hair growshalf an inch per month (of 30 days). That’smiles per hour. Enter your number as a decimal (orin scientific notation) with at least four digits beyondthe leading zeros. Use your calculator.

41.(10 pts) set8/4010s8p41.pgThis last problem is motivated by a discussion I hadwith a couple of students in our class about which for-mulas and data you should know without having tolook them up. Here is a collection of some of them.You want to be able to answer the questions in thisproblem without having to hesitate or consult a refer-ence. All of these numbers and formulas are boundto occur routinely and without advance notice in yourelementary school teaching. Of course you will beaware of most of the answers, and I hope you won’tfeel insulted. But you may be surprised by a few ofthese items. If you don’t get the correct answer rightaway look it up or figure it out, and commit it to yourmemory.

Some of your answers will be mathematical ex-pressions. For example, since the area of a rectanglewith length L and width W equals length times width,

in the first question below you need to enter LW orL*W. Use pi to denote the ratio of the circumferenceof a circle and its diameter. pi equals approximately3.14.

For a few of the numerical answers you need to en-ter numerical values, ww will not accept arithmeticexpressions.

The area of a rectangle with Length L and width Wis .

The volume of a box with Length L, width W , andheight H is .

The area of a triangle with height h and base b is.

The circumference of a circle with radius r is.

The diameter of a circle with radius r is.

The volume of a sphere of radius r is ,and its surface area is .

There are feet in a mile,feet in a yard, and inches in a foot.

There are quarts in a gallon,pints in a quart, cups in a

pint, and fluid ounces in a cup. Thusthere are fluid ounces in a gallon.

There are ounces in a pound, andpounds in a ton.

Enter the next three answers as powers of 2. Youdon’t need to work out the actual numerical value.For example, you’d say 2**5 instead of 32.

There are (not 1000) bytes in akilobyte, (not 1,000,000) bytes in amegabyte, and (not 1,000,000,000) bytesin a gigabyte.

There are days in an ordinary (not leap)year, hours in a day, minutesin an hour, and seconds in a minute.

There are seconds in an hour,seconds in a day, and hours

in a week.Working forty hours a week, and fifty weeks a year,

a full time job requires working hours peryear.

Water boils at degrees Fahrenheit andfreezes at degrees Fahrenheit. (These arethe figures for normal air pressure at sea level.)

5

The normal core body temperature of a healthy,resting adult human being is usually stated to be

degrees Fahrenheit.The number of days in the months are as follows:January: days;February: days in a leap year and days in

an ordinary year;March: days;April: days;

May: days;June: days;July: days;August: days;September: days;October: days;November: days;December: days.



Peter Alfeld.


due 7/26/06 at 11:59 PM.

Procrastination is hazardous!For some of these problems, particularly those in-

volving rational numbers and integers, you may wantto wait a few days after the set opens, until we havecovered the subject in class.

The next, and last, home work will cover the entiresemester, and serve as a review, and a preview for thefinal exam.


You buy 12 cubic yards of top soil to cover 1296square feet in your back yard. The average depth ofyour new top soil will be inches.

2.(10 pts) set9/4010s9p2.pgThis is problem B3 from section 7.3.

Write a fraction in the simplest form that is equiv-alent to each ratio:

17 to 119: / ,26 to 91: / ,97.5 to 66.3: / .3.(10 pts) set9/4010s9p3.pg

This is problem B4 from section 7.3.For each of the following pairs of ratios, enter the

letter E if the two ratios are equal, and the letter N ifthey are not equal.

5:8 and 15:257:12 and 36:604.(10 pts) set9/4010s9p4.pg

This is similar to the preceding problem.For each of the following pairs of ratios, enter the

letter E if the two ratios are equal, and the letter N ifthey are not equal.

3 : 7 and 0.6 : 1.4 ,7 : 12 and 8.4 : 14.4 ,

3 : 4 and (3+5) : (4+5) ,3 : 4 and (3×5) : (4×5) .

5.(10 pts) set9/4010s9p5.pgThis is a variation of problem B12 of section 7.3.

Three car batteries are advertised with warrantiesas follows:

Model XA: 40-month warranty, $ 34.95Model XL: 50-month warranty, $ 39.95Model XT: 60-month warranty, $ 49.95Considering only the warranties and the prices, the

best buy is Model (enter the two letter modelname), and the worst buy is Model .

6.(10 pts) set9/4010s9p6.pgYour local grocery store offers three packages ofSwiss Cheese:

10 ounces for $ 4.3. This costs cents per ounce.one pound for $ 8.96. This costs cents per

ounce.three and a half pounds for $ 30.24. This costs

cents per ounce.After due deliberation you decide to buy the kind

that tastes best.7.(10 pts) set9/4010s9p7.pg

This is problem B28 from section 7.3. Ferne, Donna,and Susan have just finished playing 3 games. Therewas only one looser in each game. Ferne lost the firstgame, Donna lost the second game, and Susan lostthe third game. After each game, the loser was re-quired to double the money of the other 2. After threerounds, each woman had $ 24. At the start, Ferne had$ , Donna had $ , and Susan had $ .

8.(10 pts) set9/4010s9p8.pgThis is problem B29 from section 7.3. A ball, whendropped from any height, bounces 1

3 of the originalheight. If the ball is dropped, bounces back up, andcontinues to bounce up and down so that it has trav-eled 106 feet when it strikes the ground the fourthtime then it was dropped from an initial height offeet.

9.(10 pts) set9/4010s9p9.pgYour family has a monthly after tax income of $ 3700.You spend 36 % of that amount for housing, 20 % fortransportation, and $ 769 for groceries. This leaves $

per month for discretionary spending.

1

10.(10 pts) set9/4010s9p10.pgThis is an extension of problem B22 from section 7.4.Shown in the following Table are data on the numberof registered motor vehicles in the United States andfuel consumption in the the United States in 1990 and1998.

Year V F1990 188.8 83521998 211.6 10,104

Here, V is the number of vehicle registrations inmillions, and F is the number of thousands of barrelsof fuel consumed each day.

Thus the number of vehicle registrations increasedby % between 1990 and and 1998. During thesame period, consumption of motor fuel went up by

%.Evidently, fuel consumption went up faster than

vehicle registration. The fuel consumption per ve-hicle increased by %.

Enter your percentages with two digits beyond thedecimal point.

11.(10 pts) set9/4010s9p11.pgSusan has $ 20.00, Sharon has $ 25.00. Susan has% less than Sharon, and Sharon has % more thanSusan.

12.(10 pts) set9/4010s9p12.pgMatch the statements defined below with the letterslabeling particular numbers. Use all the letters. Ofcourse a natural number is also a rational number, forexample. However, there is only one correct match-ing that uses all five letters A through E.

1. x is an irrational number2. x is an integer3. x is a natural number4. x is a rational number5. x is neither positive nor negativeA. x = 17

12B. x = −17C. x = 0D. x = 12E. x = π

13.(10 pts) set9/4010s9p13.pgIndicate whether the following statements are True(T) or False (F).

1. -17 is an integer2.

√49 is a rational number

3. 0 is a natural number4.

√3 is a rational number

5. π is a real number6. 3

2 is an integer7. 2 is a real number


1. The quotient of two natural numbers is al-ways a rational number

2. The product of two natural numbers is alwaysa natural number.

3. The difference of two natural numbers is al-ways an integer.

4. The quotient of two natural numbers is al-ways a natural number.

5. The sum of two natural numbers is always anatural number.

6. The ratio of two natural numbers is alwayspositive

7. The difference of two natural numbers is al-ways a natural number.


1. The quotient of two integers is always an in-teger (provided the denominator is non-zero).

2. The sum of two integers is always an integer.3. The product of two integers is always an inte-

ger.4. The difference of two integers is always an

integer.5. The quotient of two integers is always a ra-

tional number (provided the denominator isnon-zero).

6. The difference of two integers is always a nat-ural number.

7. The ratio of two integers is always positive


2

1. The difference of two rational numbers is al-ways a rational number.

2. The quotient of two rational numbers is al-ways a rational number (provided the denom-inator is non-zero).

3. The quotient of two rational numbers is al-ways a real number (provided the denomina-tor is non-zero).

4. The ratio of two rational numbers is alwayspositive

5. The difference of two rational numbers is al-ways a natural number.

6. The sum of two rational numbers is always arational number.

7. The product of two rational numbers is al-ways a rational number.


1. The difference of two real numbers is alwaysan irrational number.

2. The quotient of two real numbers is alwaysa real number (provided the denominator isnon-zero).

3. The ratio of two real numbers is never zero.4. The product of two real numbers is always a

real number.5. The difference of two real numbers is always

a real number.6. The quotient of two real numbers is always a

rational number (provided the denominator isnon-zero).

7. The sum of two real numbers is always a realnumber.

18.(10 pts) set9/4010s9p18.pg

To solve this problem you will need to set up andsolve a quadratic equation. This subject was dis-cussed in Math 1010 and Math 1050. However, forreference, here is the quadratic formula.

The solution of the quadratic equation

ax2 +bx+ c = 0

is given by

x =−b±

√b2 −4ac

2a .

I found the following definition in The HarperCollins Dictionary of Mathematics:“golden mean, golden section, or extreme and meanratio, n., the proportion of the division of a line sothat the smaller is to the larger as the larger is to thewhole, or of the sides of a rectangle so that the ra-tio of their difference to the smaller equals that of thesmaller to the larger, supposed in classical aesthetictheory to be uniquely pleasing to the eye.”Those Greek were onto something! Apparently that“extreme and mean ratio” is a specific number. Infact it is . By the way, don’t get put off bythe above piece of language which is opaque only be-cause of its brevity. The Harper Dictionary of Mathe-matics is excellent, despite occasional idiosyncrasies.It’s also an inexpensive paperback that I use almostevery day, and that I recommend highly as an addi-tion to your library.

19.(10 pts) set9/4010s9p19.pgYour European friend tells you proudly that his newcar requires only 8.1 liters of fuel to travel 100 kilo-meters. Knowing that 1 mile equals 1.609 km, and 1(US) gallon equals 3.785 liters, you compute that thismeans the car is getting miles per gallon.

20.(10 pts) set9/4010s9p20.pgYou are visiting Europe and your friend asks youwhat mileage your car is getting. You tell her that itgets 19 miles per gallon. Your friend is a math whiz,and she responds instantly: ”Hey, that means it needs

liters to travel 100 kilometers.”21.(10 pts) set9/4010s9p21.pg

This is problem B14 of section 8.1.For each of the following equations, find the inte-

ger that satisfies the equation.−x = 5, x = ,x− (−5) = −8, x = ,−5− x = −2, x = ,x+(−3) = −10, x = ,6− x = −3, x = ,x = −x, x = .22.(10 pts) set9/4010s9p22.pg

This is similar to the preceding problem.3

Solve each of the following equations.3− (3− x) = 5, x = ,3+(2− x) = −8, x = ,(1− x)− (2−2x) = −2, x = ,−(3− x) = −10, x = ,x−1 = −4−2x, x = ,x− (1− (2− (3− x))) = x, x = .

23.(10 pts) set9/4010s9p23.pgThis is problem B15 of section 8.1.

Suppose p and q are arbitrary negative integers.For the statement below, enter T if they are true forall choices of p and q, and F otherwise.

−p is negative. .p−q = q− p .−(p+q) = q− p. .−p is positive. .

24.(10 pts) set9/4010s9p24.pg

Suppose p and q are arbitrary integers. For thestatements below, enter T if they are true for allchoices of p and q, P if they are true for all positivenumbers p and q (but not all numbers p and q, N ifthe statement is true for all negative numbers p andq, and F if none of T, N, or P apply.

−(p−q) = q− p .p+q > 0 .p+q < 0 .p−q > 0 .p+q > p−q .p+q < p−q .q− (p−q) = p .q− (p−q) = 2q− p .

25.(10 pts) set9/4010s9p25.pgThe sum of two positive number is positive, the sumof two negative numbers is negative, and the sum of apositive and a negative number can be positive or neg-ative (or zero). This is summarized in the followingTable. where + indicates a positive number, - a nega-tive number, and a question mark indicates a numberof unknown sign.

add + −+ + ?− ? −

Make a similar Table for subtraction. You obtain

subtract + −+ A B− C D

where the letters A, B, C, D stand for +, -, or aquestion mark. Enter the appropriate symbols:

A: .B: .C: .D: .26.(10 pts) set9/4010s9p26.pg

This is similar to the preceding question, but your Ta-ble is for multiplication:

multiply + −+ A B− C D

Enter the appropriate symbols:A: .B: .C: .D: .27.(10 pts) set9/4010s9p27.pg

Finally, make a table for division. Assume, of course,that the divisor is non-zero.

divide + −+ A B− C D


While we are in the Table making business, make asimilar Table for the addition or subtraction of evenand odd integers. For example, the sum (or differ-ence) of two even integers is even. In the table below,enter the letter E to denote an even integer, or the let-ter O to denote an odd integer.

add E OE A BO C D

Enter the appropriate symbols:A: .

4

B: .C: .D: .29.(10 pts) set9/4010s9p29.pg

Make a similar Table for the multiplication of evenand odd integers:

multiply E OE A BO C D


The population of a hypothetical island consists ofmen, women, and children. The ratio of men towomen is 3:2. 26 % of the people are women.% are men, and % are children.

31.(10 pts) set9/4010s9p31.pgOn a neighboring island, the ratio of men to womenis 2:3. 40 % of the people are children. % aremen, and % are women.

32.(10 pts) set9/4010s9p32.pgWomen make up about 20% of the US Armed Forces.Assuming it’s exactly 20%, the ratio of men towomen in the military is . (Enter your answer asa number, or a fraction, but do not use a colon.)

33.(10 pts) set9/4010s9p33.pgThe remaining questions are from the AA Prob-lem Deck. The names of my mother, my father,my brother, my sister, and me are Robert, Annette,George, Linda, and Jean. George is younger than Iam. I am older than Jean. Annette is younger thanLinda. I am .

34.(10 pts) set9/4010s9p34.pgThe gym has 2kg and 5kg discs for weight lifting.

There are 14 discs in all. The total weight of the 2kgdiscs is the same as the total weight of the 5kg discs.The total weight of all the discs is kg.

35.(10 pts) set9/4010s9p35.pgThe L-1011 Tri Star airplane has a fuel capacity ofabout 23,000 gallons. Assuming the mileage of yourfamily car is miles per gallon this is enough fuel topower it 375,000 miles, or 15 times around the world.

36.(10 pts) set9/4010s9p36.pgYou make a 100-mil trip to the beach traveling at arate of 50 miles per hour. You make the return triptraveling at a rate of 30 miles per hour. Your averagespeed for the round trip was miles per hour.

37.(10 pts) set9/4010s9p37.pgOne fourth of the garden was filled with rose bushes.The remaining 12 square feet of the garden werefilled with mums. The area of the garden is squarefeet.

Note: Evidently this is a very small garden.38.(10 pts) set9/4010s9p38.pg

The smallest square number that is the sum of twosquare numbers is .

39.(10 pts) set9/4010s9p39.pgA dining room has 27 tables. Two thirds of the tableseach have 4 chairs. There are 6 chairs at each of theremaining tables. Thus up to people can beseated at tables in the dining room.

40.(10 pts) set9/4010s9p40.pgTim has three times as many comic books as Betsy.Betsy has two thirds as many comic books as Curt.Curt has 27 comic books. Tim has comic books.

41.(10 pts) set9/4010s9p41.pgA spider is on one corner of a cube. He wants to walkto the opposite corner of the cube. He can only walkalong the edges of the cube. If each possible path isalong three edges of the cube, there are differentpaths that the spider can take.



Peter Alfeld.


due 8/2/06 at 11:59 PM.

Procrastination is hazardous!This home work set covers a couple of new topics

(infinity, and rational exponents), but mostly it pro-vides a review of the entire semester. One of its pur-poses is to help you prepare for the final exam. Moreinformation about the final will be given in class. Re-call that the final will take place

Friday, August 4, 7:30-9:30am,in our regular classroom. It will be comprehensive,and the format will be the same as for our midtermexams.


As a warmup, solve this simple percent problem.At your 15 year high school reunion you run into

your former classmate Chester. All you rememberabout Chester is that he always struggled in his mathclasses. However, at the reunion he makes it prettyapparent that since he dropped out of high school hehas struck it rich. Finally you ask him how he madeall that money. ”Oh”, he says, ”it’s easy. I have arestaurant, and I buy steaks for 2 dollars and sell themfor 12 dollars. I live on the 10 percent.”

Actually, Chester’s profit on those steaks ispercent.

2.(10 pts) set10/4010s10p2.pgVenus orbits the Sun about 8 times every time theEarth orbits the sun 5 times. Thus 8 Venus years areabout 5 Earth years. Venus also orbits the sun aboutthree times every time Mars orbits the Sun once. As-suming these figures are exact, the length of one Marsyear is / Earth years. Enter your answers as afraction in lowest terms. (Actually, more accurately,Mars takes about 1.88 Earth years to orbit the Sun.)

3.(10 pts) set10/4010s10p3.pgYour are 5 feet 8 inches tall, and your shadow is 89inches long. A nearby tree casts a shadow that’s 53feet long. You deduce that the height of the tree isfeet.

4.(10 pts) set10/4010s10p4.pgConsider the sequence of numbers

3,4,6,8,12,14,18, . . .

The next two numbers in this sequence are and.

5.(10 pts) set10/4010s10p5.pgThe weight of an object on Mars is about 38% of itsweight on Earth. Thus an astronaut weighing 145pounds on Earth would weigh only pounds onMars. On the other hand, of course our astronautwould have to use a lot of equipment to be able tosurvive in the thin and cold atmosphere on Mars.Suppose she weighs the same with the equipment onMars as she weighs without the equipment on Earth.Thus her equipment weighs pounds on Earth. and

pounds on Mars.6.(10 pts) set10/4010s10p6.pg

Written as a Roman number, the number 459 equals. Written in the standard decimal form, the Ro-

man number CMXLIV equals .7.(10 pts) set10/4010s10p7.pg

Let S = {1,3,4,7,9} and T = {1,2,4,5,6,7}.Then S has elements and T has elements.

The set S ∪ T has elements, and S ∩ T haselements.

The set S−T has elements, and T −S haselements.

The largest element in S∪T is , and the largestelement in S∩T is .

The smallest element in S∪T is , and the small-est element in S∩T is .

8.(10 pts) set10/4010s10p8.pgA class on Mathematics for Elementary SchoolTeachers has 35 students and meets three times aweek on Mondays, Wednesdays, and Fridays. Be-cause of their schedules, one student only attendsMondays, one only attends Wednesdays, one only at-tends Fridays, 2 only attend Mondays and Fridays,3 only attend Mondays and Wednesdays, and 4 only

1

attend Wednesdays and Fridays. students attendevery day of the week. On Mondays there arestudents in the classroom, on Wednesdays there are

students, and on Fridays there are students.Of course, the teacher is in the classroom every timethe class meets, all semester long.

9.(10 pts) set10/4010s10p9.pgAccording to the US National Debt Clock on July12, 2006, just after noon, the US public debt waseight trillion, four hundred nineteen billion, one hun-dred thirty-three million, nine hundred ninety-eightthousand, four hundred fifty-four dollars. Enter thisnumber in decimal form, without spaces or commas,here: .

Compare this with the national debt on May 18,2006, which was $ 8,341,998,558,170, see Problem22 on set 2. The national debt grew byDollars since that time. Assuming that both num-bers were obtained exactly at noon, the national debtgrew by Dollars per second during the pe-riod from May 18 to July 12. (Round the last answerto the nearest dollar.)

10.(10 pts) set10/4010s10p10.pgConvert the following numbers to their decimalequivalents:

5124six = .3522six = .2215six = .

11.(10 pts) set10/4010s10p11.pgConvert the following decimal numbers to their base6 equivalents:

271 = ( )six.622 = ( )six.283 = ( )six.

12.(10 pts) set10/4010s10p12.pgIn this problem we study the closure of certain setsunder the arithmetic operations. Enter the letter C forclosed or the letter N for not closed. For example, thesum of two whole numbers is a whole number, andso the set of whole numbers is closed under addition.Thus so you want to enter the letter C for the entry Ain the Table. Of course, when examining division weonly consider non-zero divisors. So, for example, theset of real numbers is closed under division.

addition subtraction multiplication divisionwhole numbers A B C D

integers E F G Hrational numbers I J K L

real numbers M N O P

Enter your answers here:A: ,B: ,C: ,D: ,E: ,F: ,G: ,H: ,I: ,J: ,K: ,L: ,M: ,N: ,O: ,P: .

13.(10 pts) set10/4010s10p13.pgComplete the following rules. Use an asterisk todenote multiplication. I’m using capital letters forthe variables because of WeBWorK idiosyncrasies.Make sure you use capital letters in your answers aswell.

Commutative Law of addition: A+B = .Commutative Law of multiplication: A ∗ B =

.Associative Law of addition: (A+B)+C = .Associative Law of multiplication: (A ∗B) ∗C =

.Distributive Law: A∗ (B+C) = .

14.(10 pts) set10/4010s10p14.pgFill in the appropriate words.

In the equation c = a+b, c is the of a and b.c is obtained from a and b by the arithmetic operationcalled .

In the equation c = a−b, c is the of a and b.c is obtained from a and b by the arithmetic operationcalled .

2

In the equation c = a×b, c is the of a and b.c is obtained from a and b by the arithmetic operationcalled .

In the equation c = a÷ b, c is the of a andb. c is obtained from a and b by the arithmetic op-eration called . In this context, a is called the

, and b is called the .15.(10 pts) set10/4010s10p15.pg

Fill in the appropriate words.The expression

mn ,

where m and n are positive integers, is called a .m is the and n is the . If m and n have nofactors greater than 1 in common, then a

b is said to bein simplest or in lowest .

16.(10 pts) set10/4010s10p16.pgLet m = 16 and n = 19. Then the quotient of m and nis

, their product is , their sum is, and their difference is .

17.(10 pts) set10/4010s10p17.pgLet m = 11 and n = 12. Then adding m and n gives

, multiplying them gives , and dividingtheir product by their sum gives

.18.(10 pts) set10/4010s10p18.pg

The power where the exponent is the sum of a and b,and the base is the difference of a and b is .

The product of the sum and difference of a and bis .

The quotient of the sum and difference of a and bis .

If WeBWorK does not accept your answer theremay be a problem with your syntax. Use the PreviewAnswer button to see what ww thinks you are saying.

19.(10 pts) set10/4010s10p19.pgThis exercise concerns scientific notation. Fill in theblanks:0.0008153 = 8.153E .0.000000005515 = 5.515E .52170000 = 5.217E .75330000000 = 7.533E .

20.(10 pts) set10/4010s10p20.pgHere are some actually occurring numbers. Write

them in scientific notation. Remember that there isone digit before the decimal point, and it is non-zero.The land area of Earth is 57,500,000 = Esquare miles.The ocean area of Earth is 139,400,000 = Esquare miles.The length of a light year is 9,461,000,000,000,000 =

E kilometers.This equals approximately 5,880,000,000,000,000 =

E miles.The thickness of the skin of a soap bubble is approx-imately 0.0000001 = E meters. (A meter isa little more than three feet.)An ounce of Oxygen contains approximately686,000,000,000,000,000,000,000 = Emolecules.

21.(10 pts) set10/4010s10p21.pgAs part of your well developed number sense, somenumbers you want to recognize on sight as certainpowers. For example, 125 = 53, so in the first ques-tion below you enter 5 and 3. In all case you want theexponent to be greater than 1.

125 = ab where a = and b = .27 = ab where a = and b = .121 = ab where a = and b = .216 = ab where a = and b = .128 = ab where a = and b = .225 = ab where a = and b = .343 = ab where a = and b = .

22.(10 pts) set10/4010s10p22.pgWanting always to have the most up to date comput-ing equipment, you upgrade your computer to one ofthe new 1 teraByte hard disks. The manufacturer con-siders 1 teraByte to be 1012 bytes (1,000 gigaBytes).Your computer, however, thinks binarily and consid-ers a teraByte to be 240 bytes. So it tells you thatyour new Disk Drive has a capacity of teraBytes.That’s a difference of percent. (Enter your an-swers as decimals with at least 3 digits beyond thedecimal point.)

You consider organizing a nationwide campaign tointroduce truth in advertising to the computer indus-try. On second thought, you decide to explore withyour students whether or not, and if so, why, that dis-crepancy gets larger as disk drives get larger.

3

23.(10 pts) set10/4010s10p23.pgThis is a simple review of the standard algorithm foraddition. Fill in the variables as required. All an-swers are single digit whole numbers. If you wouldordinarily leave a location blank enter the digit 0.The columns in this problem are set apart from eachother for increased clarity, but of course we are sim-ply adding the two three digit numbers 539 and 785.


+ 7 8 5− − − −c d e f

,

a = ,b = ,c = ,d = ,e = ,f = .24.(10 pts) set10/4010s10p24.pg

Use the Washington algorithm for the subtraction812−187.

8 1 2− 1 8 7

a b− − −c d e

wherea = ,b = ,c = ,d = , ande = .25.(10 pts) set10/4010s10p25.pg

Fill in the digits in the following calculation:

7 8× 2 9− − −a b c

d e f− − − −g h i j

We get

a = ,b = ,c = ,d = ,e = ,f = ,g = ,h = ,u = , andj = .

26.(10 pts) set10/4010s10p26.pgFill in the digits in the following division with re-mainder:

a b− − −

2 7 | 6 9 1c d− − −e f gh i j− − −

k lHere,a = ,b = ,c = ,d = ,e = ,f = ,g = ,h = ,i = ,j = ,k = ,l = ,

27.(10 pts) set10/4010s10p27.pgLet (p,q) be the prime twin with q > p > 110 whereq is otherwise as small as possible.

p = , andq = .

28.(10 pts) set10/4010s10p28.pgThe prime factorization of 4641 is

4641 = × × ×4

where the prime factors are sorted by increasingsize.

29.(10 pts) set10/4010s10p29.pgFind the greatest common factors of the pairs of num-bers below. For the more complicated examples, Irecommend you use the Euclidean Algorithm and acalculator.

gcf(533,2911) =gcf(1219,901) =gcf(3551,2077) =

30.(10 pts) set10/4010s10p30.pgSuppose in the game of Nim your opponent starts byremoving 2 matches from the fourth row.

1. |2. | |3. | | |4. | |5. | | | | |

There is just one winning move. You removematches from row .


45 − 3

8127 + 10


ab

wherea=

and b= .32.(10 pts) set10/4010s10p32.pg

This is a fun problem to think about when it’s a hun-dred degrees outside.

It takes you 6 hours to shovel snow on your prop-erty. It takes your brother 3 hours to do the same job.Working together it takes you and your brother a/bhours to shovel that snow, where a = and b =

. (Express your answer in simplest form.)33.(10 pts) set10/4010s10p33.pg

Let

z =

−23−45−67

Then z = / .

Enter your answer as a rational number in simplestform.

34.(10 pts) set10/4010s10p34.pgEnter numerical values for the following powers withrational exponents:

811/2 = ,811/4 = ,1285/7 = ,128−5/7 = / .In the last case, enter your answer as a fraction in

lowest terms.35.(10 pts) set10/4010s10p35.pg

A certain population of blue bellied bumble bees dou-bled its size in 12 years. Thus its average annualgrowth rate was percent.

36.(10 pts) set10/4010s10p36.pgWritten as a fraction in simplest form,

7.407 = / .

37.(10 pts) set10/4010s10p37.pgThe repeating decimal form of the fraction 17/33 hasa period and a repetend .

38.(10 pts) set10/4010s10p38.pgAmong the students in your school, 170 are boys, and181 are girls. Thus the ratio of boys to girls isand the ratio of girls to boys is .

Put differently, the fraction of students who areboys is and that of students who are girls is .

Put yet differently, percent of the students aregirls, and percent of the students are boys.

39.(10 pts) set10/4010s10p39.pgIndicate whether the following statements are True(T) or False (F). (For all statements involving divi-sion, it is understood that the denominator is non-zero.)

1. The set of integers is closed under addition,subtraction, and division.

2. The difference of two natural numbers mightnot be rational.

3. The quotient of two rational numbers may benon-rational.

4. All rational numbers are real, but some realnumbers aren’t rational.

5. The square root of 49 is irrational.6. Not all rational numbers are integer.

5

7. All integers are rational.


1. The square of a real number is always greaterthan that number.

2. The square root of a positive rational numbermay be irrational.

3. The square root of 51 is irrational.4. The square of a rational number is rational.5. The square of an integer is always integer.6. A negative number is always less than a pos-

itive number.7. The square of any real number is non-

negative.

41.(10 pts) set10/4010s10p41.pgAs we discussed in class, we say that a set A is smallerthan a set B if there is a 1-1 correspondence betweenA and a subset of B, but no such correspondence be-tween B and a subset of of A

Indicate whether the following statements are True(T) or False (F).

1. The set of rational numbers is larger than theset of integers.

2. The set of integers is larger than the set ofprime numbers.

3. The set of real numbers is larger than the setof real numbers between 3 and 4.

4. The set of integers is larger than the set ofeven integers.

5. The set of real numbers is larger than the setof rational numbers.

6. For any infinite set, there is a larger set.7. Between any two distinct rational numbers

there are infinitely many rational numbers..

42.(10 pts) set10/4010s10p42.pgIt’s been a long haul, and you have reached the last

WeBWorK problem, not just in this set, but in this en-tire semester! What will you do without these weeklyww problems? As the grand finale, let me pose twopuzzles. Send me your answer by email. I will for-ward the best answers to the whole class, with yourname attached.

Here is the first puzzle. A surgeon wears rubbergloves to protect both the patient and the surgeon.Suppose you need to perform three operations (ondifferent patients), but you have only two pairs ofgloves. Each surgery requires the use of both hands.How do you proceed? (Disclaimer: this is a theoreti-cal problem, don’t try this at home. Myself, the MathDepartment, and the University of Utah, will rejectany and all liability for your attempts to save on rub-ber gloves!)

I have been told the second puzzle used to be posedin the Physics Department of a certain University intheir Ph.D. qualifying exams. A candidate wouldhave five minutes to figure out the answer, and wouldfail the exam if unsuccessful. Of course, you are not(yet!) a Ph.D. candidate in physics, but you do havemuch more time than five minutes, and the problemrequires only common sense for its solution. Youhave a supply of fuses and matches, as many as youlike of each. Once you light a fuse at one end it willburn for exactly one hour. However, on each fuse, theflame moves at a different and non-constant speed.So, for example, to measure half an hour, it would notwork to cut the fuse in the middle and then light onehalf. Such a half fuse might burn any amount of timebetween 0 and 60 minutes. Describe how you can useyour fuses and matches to measure a time interval of45 minutes. (If you want to turn this into a somewhathigher math problem, ask yourself just what kind oftime intervals can be measured with those fuses andmatches.)

To get credit for this ww question, type the sen-tence ”I read this”, without the quotation marks, here:

.


the one point contest - home - math · 2006. 8. 10. · help you familiarize yourself with ww. here...

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