Here is a collection of problems that I have asked in previous semesters on the material to be covered on final exam that was not covered previously. Obviously you will also want to work on the two sets of sample questions posted previously, because the final exam covers the entire course. (Some of the questions listed here also deal with earlier material.) I think you will learn a lot by working on these problems, and it will prepare you well for the test.
(Lots of problems on the actual test will look like these, with some changes, of course.) Many of them require you to think about something you have not seen before—those are certainly parts of what learning mathematics and becoming a critical thinker entail. The instructions are typically as follows:
Work all 30 problems and clearly state the answers to the questions being asked. Except where noted, show your work and provide full explanations; answers without explanation receive little credit. Remember that you need to communicate effectively with your reader.
Point values are indicated; the total is 220, but the test will be graded as if the total were 200, so you get 20 points of errors for free. No books or notes are allowed; a calcula- tor (but no other electronic devices) may be used (but only for calculation—not to store information you are supposed to know in your head). All numerals written on this test are in base ten unless otherwise stated. Everything means exactly what it says; ask me if anything seems unclear. You can “buy” a hint for a few points deducted if you are unable to make progress on a question.
1. The following is from a recent magazine column by Marilyn vos Savant, who advertises herself as the world’s smartest woman. She often presents readers with a mathematical or logical puzzle.
Assume that the following statements are true. If Ashley goes to the concert, Brandon will go. If Brandon goes to the concert, Chelsea will go. If Chelsea goes to the concert, Daniel will go. Only two of the four students went to the concert. Who were they?
Solve her puzzle, explaining the reasoning behind your answer.
2. Think about 11/56 and 10/51 on the number line.
(a) Which is to the right and which is to the left? Why?
(b) Find a rational number between them.
(c) Find an irrational number between them.
3. This problem deals with the issue of if and when a percentage change greater than 100% might make sense. In each case either explain with a real-world example how the situation can occur, or explain why this can never happen.
(a) Can a quantity increase more than 100%?
(b) Can a quantity decrease more than 100%?
4. Consider the number x = 12.158 = 12.158585858 . . . .
(a) Explain completely, using words, math symbols, and/or diagrams, what this no- tation really means. How does one make sense out of this infinite decimal ex- pansion? How do we get a handle on what number we are talking about here?
I’m not asking what the bar means—this is a question about the meaning of the mathematical idea, not about the notation. [HINT: It is not equal to, and does not get close to, a terminating decimal.]
(b) If this number x is rational, express it as ab for whole numbers a and b (obviously, you need to show how, not just use a calculator button). If it is irrational, explain clearly why it cannot be expressed in this way.
5. One of the themes that I kept stressing and preaching in MTE 210, which is not explicitly one of the NCTM “standards”, is that mathematics deals with statements that have meaning, that it is not just a set of facts or procedures to be memorized.
Relate this to two of the NCTM “process” standards in a couple of sentences each (one could give correct answers with any two of the five):
(a) NCTM Standard:
(b) NCTM Standard:
6. Let P be the set of the people in the world (you, me, Barack Obama, . . . —all 6.8 billion of us). Let C be the set of countries of the world (China, India, USA, . . . ).
(a) Give an example of a function f from P to C; that is, P is to be the set of inputs, and C is to be the set of outputs. State your answer in the form “f (x) = . . .”;
you fill in the dots with a nice rule specifying your function (what the output is for input x). In addition, draw a schematic diagram showing a portion of the domain and range (ovals), together with a few typical arrows for your function.
(b) Give an example of a function g from C to P ; that is, C is to be the set of inputs, and P is to be the set of outputs. Again, state your answer in the form
“g(x) = . . .” and draw a diagram showing a portion of the domain and range, together with a few typical arrows for your function.
(c) One of f (g(you)) and g(f (you)) makes sense, and one does not. Using your answers to parts (a) and (b), compute the one that makes sense. [Here “you”
means you—the person sitting in your seat taking this test.]
7. Let S be the set of students at Oakland University, and recall that W is the set of whole numbers.
(a) What do we mean by “a function from S to W ”? I’m not looking for a specific example here—I want you to tell me the definition of what a function is. The amount of space here gives some indication of how long and detailed your answer should be. (One good response I can think of uses about 14 words.)
(b) Give an example of a function f of some personal, social, or educational sig- nificance from S to W ; that is, S is to be the domain (set of inputs), and W
is to be the codomain (set of possible outputs). State your answer in the form
“f (x) = . . .”; you fill in the dots specifying what the output is for input x.
(c) Draw a schematic diagram with ovals (for the domain and codomain) and arrows showing a portion of how your function operates.
8. Proportional reasoning (using proportions and ratios to solve problems, what was called “the rule of three” a hundred years ago) is an important topic in elementary and middle school mathematics. (In fact, the NCTM “Number and Operations” standard for grades 6–8 states that “all students should develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios”; and the NCTM has singled it out as one of three “focal points” for grade seven mathematics.) Make up a good, realistic story problem that can be solved using proportional reasoning, and then solve it. Make sure to state the question clearly (it should be worthy of being in a textbook or on a standardized test), and make sure to communicate clearly as you write up the solution.
9. Make up a good, realistic story problem whose solution involves performing a division of whole numbers (specifically, the calculation 38 ÷ 5), where it’s the remainder in this division (rather than the quotient) that is the answer to the question. Then give a clear and complete solution to the problem.
10. Consider the “absolute difference” operation, which for this exercise we will denote by . It is defined by a b = |a − b|.
(a) Explain why this operation is or is not commutative.
(b) Explain why this operation is or is not associative.
(c) Explain why the set of integers is or is not closed under this operation.
11. We gave careful proofs in class of two things: that there are infinitely many prime numbers, and that the √
2 (square root of 2) is irrational. Pick one of these and give the proof on this page. Make sure to communicate convincingly. Your proof should have lots of words in it, as well as any necessary mathematical calculations.
12. A story in The Detroit Free Press on January 18, 2009 about a Florida beach resort read in part, “Suddenly sleepy Caladesi’s attendance skyrocketed nearly 30%. It at- tracted 100,000 ferry visitors in 2008, up from about 70,000 the year before.” Explain how the writer got this figure of 30% from the data given in the second sentence, and explain clearly why it is incorrect. Make sure to include a computation of the correct percentage increase (reported to an appropriate degree of precision) as part of your explanation. (Incidentally, the newspaper published a correction the following day.)
13. Consider the following statement about whole numbers:
If n is a multiple of both 2 and 6, then n is a multiple of 12.
We are interested in whether this statement is true for all whole numbers n.
(a) Write the negation of this statement (what it would mean for it to be false).
[Remember that the negation of an “if” statement is not an “if” statement.]
(b) Either explain why the original statement is true for all n, or else provide a counterexample (which will be a value of n that makes the negation—the correct answer to (a)—true).
14. Let A be the set of rational numbers between 5 and 6, inclusive, and let B be the set of irrational numbers between 5 and 6.
(a) Write down one element of A.
(b) Write down one element of B.
Now pick one of the following set operations to use for parts (c) and (d): either intersection (∩) or union (∪). CIRCLE YOUR CHOICE.
(c) Define your operation applied to the sets A and B. (I don’t want to know the answer here; I want the definition—what it means.)
(d) What is A ∩ B or A ∪ B here (answer for whichever one you chose)? The answer is very simple in each case.
15. One of my MTE 210 students in a previous semester had a daughter in the fourth grade. The daughter had a math quiz, one question of which was to fill in the blank:
6 ÷ = 0. The girl wrote 6 in the blank, and the teacher marked it wrong; the correct answer, the teacher said, is 0. Who is right? Why? Explain completely. [This story is completely true.]
16. We talked a lot in this course about the problem-solving process.
(a) Name three different problem-solving strategies that we have talked about in this course.
(b) Name two different things one might want to do during the “Looking Back” stage (Step 4) in the problem-solving process.
17. How has your attitude toward mathematics changed as a result of taking MTE 210 this semester? Think about your feelings about what mathematics is, how it applies to you, what is important in doing, learning, and teaching mathematics, and misconceptions you previously had. In your answer you must refer explicitly to at least one principle, one content standard , and one process standard from the NCTM’s Principles and Standards for School Mathematics. Please underline those three words or phrases in your essay. Although I’ve allowed an entire page for your essay, please do not feel compelled to fill the page.
18. For this problem, consider the operation “averaged with”, which for this problem we’ll denote by the symbol @. For example, 3 @ 11 = 7, because the average of 3 and 11 is (3 + 11)/2 = 7.
(a) Is this operation distributive over itself? Write down the relevant equation that expresses this property. Then try an example and state what your example leads you to conclude or conjecture.
(b) Is the set of negative rational numbers closed under this operation? Explain why.
19. My house number is 3125. This is of course a numeral in base ten, and I’m proud of the fact this happens to be 55.
(a) If I had the same numeral on my house and we all used base six, then my house number would be a prime number. Write this number in base ten.
(b) If my house number was 55 and we all used base six, then the numeral written on my house would have five digits. Find this base-six numeral.
20. We dealt with logic (statements, together with connectives like “if. . . then” or “or”) as a mathematical system. The distributive property of the negation (“not”) operation over the conjunction (“and”) operation would say
∼(p ∧ q) is logically equivalent to (∼p) ∧ (∼q).
(a) An application of this that I heard many years ago involved a lifeguard at a swimming pool announcing conditions under which children could use the “baby pool”. She said the child must be under six years old and be supervised by a parent. Apparently some children were violating this rule (in other words, the negation of this condition was happening). She summarized this bad situation by invoking this distributive law. Let p = “the child is under six years old” and q = “the child is supervised by a parent”. The left-hand side of the equivalence displayed above would then be “it is not true that both the child is under six years old and the child is supervised by a parent.” Write down in English the right-hand side of this equivalence (this is what the lifeguard said was happening).
(b) In fact, this distributive law is not correct. Instead, De Morgan’s Law tells us the correct way to negate a conjunction. In symbols, what does De Morgan’s Law tell us that ∼(p ∧ q) is logically equivalent to?
(c) Write out the correct negation (the correct answer to part (b)) in words. This is what the lifeguard should have said when talking about how the rules were being violated.
21. In one of the following scenarios it is appropriate to set up a simple proportion using the given numbers in order to solve the problem. In the other two, it is not that straightforward. Solve the problem for which it is appropriate, stating the answer with an appropriate degree of precision.
• A piece of solid wood in the shape of a cube having side length 3 inches weighs 568 grams. A second piece of the same type of wood also in the shape of a solid cube, has side length 5 inches. How much does the second piece weigh?
• The normal high temperature for Detroit in April is 58◦F (Fahrenheit), which is 14◦C (Celsius). The normal high temperature for Detroit in May is 68◦F.
Approximately what is the normal high temperature for Detroit in May on the Celsius scale?
• Aleisha has saved pennies and has 159 of them; they weigh a total of 14.0 ounces.
Maurice has 201 pennies. Approximately what is the weight of Maurice’s pennies?
22. In one of the following scenarios it is appropriate to set up a proportion in order to solve the problem. In the other, it is not. Solve the problem for which it is appropriate.
• Two cars travel at the same constant speed on the same road. Aleisha drives for 75 minutes and goes 50 miles. Maurice goes 60 miles. For how many minutes did Maurice drive?
• Aleisha got a total of 570 points during the semester in her math class, and according to the professor’s grading system this gave her a 3.8 grade in the course.
Maurice got 450 points. What grade did Maurice get?
23. Suppose that 2% of all baseball players use steroids. Suppose also that there is a test for steroid use that gives false positives 4% of the time (that is, when a person who does not use steroids takes the test, 96% of the time the test will confirm that he does not use steroids), and gives false negatives 3% of the time (that is, when a person who does use steroids takes the test, 97% of the time the test will confirm that he does use steroids). If a random player is tested and the test indicates steroid use, how likely is it that the player actually uses steroids? State your answer as a percentage, rounded to an appropriate degree of precision. [HINT: Set up a table similar to what we did in class with the test for the rare disease.]
24. Consider the following four scenarios involving a shop-keeper during the holiday sea- son, which relate to the multiplication of positive and negative numbers. For each one, write down in the space provided the multiplication fact that yields the answer to the question of how much extra profit the shop-keeper will earn. Keep in mind that getting less profit can be viewed as getting a negative amount of extra profit. Profit comes from net sales minus expenses (like paying the staff, electricity for the lights, and so on ).
(a) Business is booming, and the shop-keeper figures that she makes a $300 profit for every hour the shop is open. So she decides to stay open 2 hours longer than normal. How much extra profit will result from doing this?
× =
(b) Business is booming, and the shop-keeper figures that she makes a $300 profit for every hour the shop is open. However, she has a family emergency and has to
close 2 hours earlier than normal. How much extra profit will result from doing this?
× =
(c) Business is pretty poor this year, and the shop-keeper figures that she loses $300 for every hour the shop is open. Nevertheless, it’s the holiday season, so as a service to her customers she decides to stay open 2 hours longer than normal.
How much extra profit will result from doing this?
× =
(d) Business is pretty poor this year, and the shop-keeper figures that she loses $300 for every hour the shop is open. Therefore she decides to close 2 hours earlier than normal. How much extra profit will result from doing this?
× =
25. Is the fraction shown below in simplest terms? If so, explain why. If not, find its simplest form. You need to show how one calculates this; using a button on a calculator that does the bulk of the work automatically is not sufficient.
37901 53303
26. Suppose there are three sets, A, B, and C, such that A and B are not disjoint and neither of them is a subset of the other, and furthermore C ⊆ A ∩ B.
(a) Draw a Venn diagram showing these three sets that is consistent with the given information. [HINT: Draw A and B first.]
(b) In your diagram, shade A − C.
27. A story in The Oakland Post on November 19, 2008, concerned the experiences and behaviors of OU students. It was reported in that article that 42.3% of sexually active women polled had never been tested for STIs.
(a) If 41 sexually active women polled had never been tested for STIs, then how many sexually active women were polled?
(b) What is the point of reporting this number as a percentage, rather than just reporting the raw data (“41 out of [correct answer to part (a)] sexually active women polled had never been tested for STIs”)?
(c) Is the degree of precision used in this story appropriate, or should the writer have used more precision (including more decimal places) or less precision? Why?
28. Come up with a good divisibility rule for 12. In other words, if I were to give you a large number, written in our usual base-ten place-value system, such as 123,456,780, and ask you to determine whether or not it is divisible by 12 just by looking at the digits of the numeral, without actually carrying out the division, how could you do it? Make sure you state your rule clearly and unambiguously, in the form “A whole number is divisible by 12 if and only if . . . ”. You do not have to explain why your rule works.
I don’t want to know whether the particular number mentioned above is or is not divisible by 12; I want a general rule.
29. Bustin Geebers gave a series of concerts, one each night for several weeks. Word got out that he was very entertaining, so the attendance increased with each performance.
The first show drew 120 people. After that, each show had 21 more people in the audience than the previous show. On the final night of his concert series, the audience numbered 687.
(a) How many performances did Bustin give? You must solve this problem in a manner that doesn’t involve something like laboriously adding 21 until you reach 687.
(b) What was the total attendance for the whole series? You must solve this problem in a manner that doesn’t involve adding up all the attendance figures.
30. We talked about the prime numbers as the multiplicative building blocks of the whole numbers.
(a) What is a prime number? State the definition precisely.
(b) According to my fancy calculator, 10110 = 110,462,212,541,120,451,001. Is this a prime number? Why or why not?
31. This problem (for which I just made up fictitious data) deals with people identifying themselves as Democrats or Republicans, and as conservative or liberal in their outlook on social issues. Suppose that 45% of all people are Democrats and the rest are Republicans. Suppose that 70% of the Democrats are liberal and the rest conservative.
Suppose that 10% of the Republicans are liberal and the rest conservative. What fraction of conservatives are Republicans? State your answer as a percentage, rounded to an appropriate degree of precision. [HINT: Set up a table similar to what we did in class with the test for the rare disease.]
32. Consider the operation “minus the reciprocal of”, which for this exercise we will denote by ♣. Thus, for example, 3 ♣ 5 = 2.8 (because the reciprocal of 5 is 15, or 0.2, and 3 − 0.2 = 2.8).
(a) Write down the equation that expresses the statement that this operation, ♣, is distributive over addition. Use a, b, and c to stand for numbers.
(b) Explain why this operation, ♣, is or is not distributive over addition.
(c) Explain why the set of positive rational numbers is or is not closed under this operation, ♣.
33. In May of 2012, voters in Clarkston defeated a $20 million bond proposal. The Oakland Press reported that 66.15% of the voters voted no.
(a) The actual results were that 5755 people voted against the proposal. How many people voted in favor of it?
(b) What is the point of reporting this number as a percentage, rather than just reporting the raw data (“5755 voted against the proposal and [the correct answer to (a)] voted for it”)?
(c) Is the degree of precision used in this story appropriate, or should the writer have used more precision (included more decimal places) or less precision? Why?
34. A magazine columnist, Marilyn vos Savant (who claims she is the smartest woman in the world), posed the following brainteaser: A woman, age 45, has two daughters, ages 20 and 25. One year from now, the sum of the daughters’ ages (26 + 21 = 47) will exceed their mother’s age (46) by one year. Two years from now, the sum of the daughters’ ages (27 + 22 = 49) will exceed their mother’s age (47) by two years. The gap between the sum of the daughters’ ages and the mother’s age keeps increasing steadily. When will the sum of the daughter’s ages be double their mother’s age? (As is customary, we measure ages only in whole numbers.)
(a) Solve the brainteaser. (There are at least two or three good ways to do this.) (b) Marilyn prefaces her solution with “Surprisingly, one needs no mathematics to
solve this problem.” Even without knowing what her solution is, and even if you were not able to do part (a), comment on Marilyn’s remark, in light of what you have learned in this course. Two or three sentences should suffice.
35. The following column by Marilyn vos Savant (who claims she is the smartest woman in the world) appeared in Parade magazine several days ago. She stated the following math problem: “Three people decide to start with $10,000 each and compete to see who makes the most in the stock market over a period of three years. The first person gains 20 percent the first year, then 3 percent the second year, then 2 percent the third year. The second person gains 2 percent, then 3 percent, then 20 percent. The third gains 20 percent, then 2 percent, then 3 percent. Which person gained the most after three years?” After stating the answer (without providing a reason for why it is correct), she wrote that if we don’t believe it, we should check it with a calculator.
Explain what the answer is (i.e., which person—if any—gained the most) and why that answer is correct without using a calculator—in fact with doing absolutely no numerical calculations at all. [Note that you are not being asked for the amount of the gain—just to figure out which person gained the most. A good response here will mention some important properties of one or more arithmetical operations. HINT:
Adding the percentages is not relevant or appropriate here—it is not true that each person gained 25%.]
36. For each part, simplify as much as possible, or explain completely why the fraction cannot be simplified. Relying on a calculator to provide the answer will earn no credit.
(a) 7324− 7322
7323 (b) 2360 + 1
2160
37. When I was in junior high school, my friends and I used to like to go bowling. In those days, you had to keep your own score (nowadays it is all done automatically at most bowling alleys). For fun and to practice our math, we kept score using base five (yes, we were taught about bases other than ten, in the seventh grade as I recall). A perfect game in bowling is to get twelve strikes in a row, resulting in a score of 300 (using base ten numerals).
(a) If my score was recorded as 300 using base five, what was my score really (i.e., written in base ten)?
(b) What would the score of a perfect game be if written in base five?
38. In ancient Greece, the citizens sometimes had referenda on various issues, and these were carried out by having each voter put a shard (a small piece of broken pottery) into a large urn. If you wanted to vote yes, then you put in a white shard, and if you wanted to vote no, then you put in a black shard. Assume that the shards varied in size, that there was never any question about what color one was, and that they never broke into smaller pieces.
(a) Explain how the vote-counters could tell whether or not a proposal passed, without having to count or use arithmetic or numbers in any way. (A simple majority was required for the proposal to pass—i.e., more than half the votes had to be in favor.) [HINT: Think about our definition of equivalent sets.]
(b) Now answer the same question but assume that a two-thirds majority vote was required—i.e., at least two thirds of the voters must have put in a white shard in order for the proposal to pass. Again, your method must not use counting or arithmetic.
Multiple choice. Each of the following problems is worth 5 points for the correct an- swer, 0 points for an incorrect or no answer (so guess any you don’t know). No work or justification need be shown for this section; just circle the letter of the one best answer.
Make sure to read all the choices before deciding.
39. Let n be the 16-digit whole number all of whose digits are 2s. Which one of these is a divisor of n? (To be explicit, n = 2,222,222,222,222,222.)
(a) 0 (b) 3 (c) 4 (d) 11 (e) 222
40. Consider these two fractions involving Oakland University students: (1) the percent- age of elementary education majors at OU who are female; and (2) the percentage of female students at OU who are elementary education majors. What is the relationship between them?
(a) They are the same thing.
(b) They are reciprocals of each other.
(c) Their sum has to be 100%.
(d) Certainly (1) is smaller than (2).
(e) None of the above.
41. Goldbach’s conjecture states that every even number greater than 2 can be written as the sum of two prime numbers (for example, 100 = 11 + 89 and 6 = 3 + 3). No one knows whether this statement is true or not—no one has been able to prove that it has to be true for every even number, and no one has found a counterexample to show that it is false. What would a counterexample consist of?
(a) An even number greater than 2 that can be written as the sum of two prime numbers.
(b) An even number greater than 2 that cannot be written as the sum of two prime numbers.
(c) An odd number greater than 2 that can be written as the sum of two prime numbers.
(d) An odd number greater than 2 that cannot be written as the sum of two prime numbers.
(e) A proof that no even number greater than 2 can be written as the sum of two prime numbers.
42. Our divisibility test for 6 in base ten is that 6 | n if and only if 2 | n and 3 | n (and then we had quick tricks for each of these last two). When is a whole number written as a base six numeral divisible by 6?
(a) if and only if the sum of its digits is divisible by 3 and its last digit is 0, 2, 4, 6, or 8
(b) if and only if the sum of its digits is divisible by 3 and its last digit is 0, 2, or 4 (c) if and only if the sum of its digits is divisible by 6
(d) if and only if its last digit is 0, 2, or 4 (e) if and only if its last digit is 0
43. We proved two theorems in this class: that there are infinitely many prime numbers, and that √
2 is an irrational number. Which one of the following statements is true?
(a) We showed that √
2 is an irrational number by looking at its decimal expansion (1.414213562. . . ) and noticing that it had no pattern.
(b) We showed that√
2 = 22619537/15994428, which is irrational because its decimal expansion doesn’t repeat a block of digits forever.
(c) Even though we proved these statements, each of them is only a theory, like evolution or intelligent design, and we will never know for sure whether they are true.
(d) Using the Fundamental Theorem of Arithmetic, we showed that given any finite set of prime numbers, there had to be another prime number not in that set.
(e) We showed that there are infinitely many prime numbers by proving that 2n− 1 is a prime number whenever n is a prime number (for example, 27− 1 = 127 is prime and 25− 1 = 31 is prime).
44. In golf, you need to add up 18 whole numbers, all of which (if you’re not too skilled at the game) are somewhere around 5, in order to calculate your final score. In doing this, you might look for pairs or triples of numbers that add up to 10, such as 3 + 7 or 4 + 6 or 3 + 3 + 4, and compute their sums first. What property or properties of the whole numbers make this a valid computational technique? [Hint: Answer the question being asked—“why is this valid”, not“why is this efficient”.]
(a) the distributive law of multiplication over addition (b) the commutative and associative laws for addition
(c) the commutative and associative laws for multiplication (d) the fact that we use BASE TEN
(e) the fact that 10 is not a prime number
45. In golf, you need to add up 18 whole numbers, all of which (if you’re not too skilled at the game) are somewhere around 5, in order to calculate your final score. In doing this, you might look for pairs or triples of numbers that add up to 10, such as 3 + 7 or 4 + 6 or 3 + 3 + 4, and compute their sums first. What property or properties of the whole numbers make this an efficient computational technique? [Hint: Answer the question being asked—“why is this efficient”, not“why is this valid”.]
(a) the distributive law of multiplication over addition (b) the commutative and associative laws for addition
(c) the commutative and associative laws for multiplication (d) the fact that we use BASE TEN
(e) the fact that 10 is not a prime number
46. “On the third day of Christmas, my true love gave to me: 3 French hens, 2 turtle doves, and 1 partridge,” for a total of 6 items. Not content to let Christmas last just twelve days, my true love extended it for almost a whole year! If she followed the same pattern, how many items did she give me on the 363rd day of Christmas?
(a) 363 (b) 364 (c) 65,703 (d) 66,066 (e) 132,132
47. In the decimal numeral 12345.06789, what digit is in the ten-thousandths place?
(a) 1 (b) 2 (c) 7 (d) 8 (e) 9
48. Suppose that ab is a proper fraction, where a and b are positive integers. Suppose we add the same positive integer n to both numerator and denominator. Call the result x. How does x compare to ab?
(a) x = a
b (b) x < a
b (c) x > a
b (d) x = a
b + n (e) x = a b + n
n 49. Which of the following statements from our discussion of number theory is not correct?
(a) The prime numbers form the building blocks for the whole numbers as a multi- plicative structure.
(b) An efficient way to find the greatest common divisor of two very large whole numbers is the prime factorization method.
(c) It can happen that the least common multiple of two whole numbers is the product of those two numbers, even if the numbers are not prime.
(d) Prime numbers were interesting to the ancient Greeks, and now they form the basis of secret coding systems used in Internet commerce.
(e) 467932243 · 21611 6= 67 · 1709 · 88316321 [You can assume that the five numbers shown here are prime numbers.]
50. There is no reason that we cannot write “decimals” in base six, extending the notation system in a manner analogous to what we did in base ten, but remembering that our grouping quantity is six, not ten. What is the value of 12.3six?
(a) nine
(b) twelve and one third (c) twelve and three sixes (d) eight and one half
(e) eight and three tenths
51. Which of the following is not something that the subtraction operation is good for?
(a) making sense of the partitive model
(b) telling how much bigger one number is than another number (c) finding a missing addend in an addition statement
(d) finding out by what amount a quantity has changed (e) telling how far apart two numbers are on the number line
52. If A and B are sets such that n(A) = 42 and n(B) = 17, what is the greatest possible value of n(A − B)? [Recall that A − B = {x | x ∈ A ∧ x /∈ B} and that n(S) is the cardinality of (the number of elements in) the set S.]
(a) 0 (b) 17 (c) 25 (d) 42 (e) 59
53. Which of the following statements about whole numbers, rational numbers, decimals, and fractions is always true? [Be careful here. Some of these statements are false for somewhat subtle reasons.]
(a) Our definition of “rational number” is that it is a number whose decimal rep- resentation either terminates or, from some point on, repeats the same block of digits forever.
(b) If the diameter of a circle is a whole number, then its circumference is an irrational number.
(c) If a and b are positive whole numbers with no common factors greater than 1, and b has a prime factor of 2 or 5, then the decimal representation of a
b terminates.
(d) The decimal representation of 7
29 goes on forever without any pattern.
(e) If a is a whole number, then the decimal representations of a
15 does not terminate.
54. Suppose you are estimating a subtraction calculation a − b, where a and b are large whole numbers, with a > b, and you round both a and b up. What can you conclude about your estimate compared to the exact answer?
(a) Your estimate will necessarily be smaller than the true answer.
(b) Your estimate will necessarily be larger than the true answer.
(c) Your estimate will be larger than the true answer if and only if you added more to a than you did to b.
(d) Your estimate will be larger than the true answer if and only if you added more to b than you did to a.
(e) Your estimate will be larger than the estimate you would get by rounding both a and b down.
55. When using the chip (charged field) model for representing integers, if the circle rep- resenting a certain number x contains some black chips (positive charges) and some red chips (negative charges), which one of the following is necessarily |x| (the absolute value of x)?
(a) the number of black chips (b) the total number of chips
(c) the number of black chips minus the number of red chips (d) the number of red chips minus the number of black chips
(e) the number of chips that would remain in the circle if we removed as many pairs of oppositely colored chips as possible
56. Which one of the following is a correct way to model or define division? Assume that a and b are whole numbers, that b 6= 0, and that b | a (a is a multiple of b).
(a) a ÷ b tells us the number of parts there will be if a objects are split into b parts of equal size.
(b) a ÷ b tells us how big each part will be if a objects are split into b parts of equal size.
(c) a ÷ b is the unique number c for which b = a · c.
(d) a ÷ b tells us the number of times we have to subtract b, starting at a, until we reach a number less than 0.
(e) a ÷ b tells us how far apart a and b lie on the number line.
57. Consider the fraction 11111111111
41943040000. How many digits after the decimal place are in the simplest decimal for this number? You will save some time here if I point out the fact that the prime factorization of the denominator is 226· 54.
(a) 4 (b) 12 (c) 26 (d) 30 (e) 104 58. What is the Fundamental Law of Fractions?
(a) the statement that “canceling” equal terms from the numerator and denominator of a fraction gives an equivalent fraction
(b) the statement that “canceling” equal factors from the numerator and denominator of a fraction gives an equivalent fraction
(c) the statement that fractions are multiplied by multiplying numerators and mul- tiplying denominators
(d) the statement that fractions are divided by multiplying the dividend by the re- ciprocal of the divisor
(e) the statement that not all fractions have decimal equivalents
59. Which one of the following statements is true about the repeating decimal 2.49, which means 2.4999 . . .?
(a) It equals 2.5.
(b) It is slightly smaller than 2.5.
(c) It equals 3.
(d) It does not represent a number.
(e) It represents an irrational number.
60. Which of the following is a correct statement?
(a) 7 ÷ 0 is undefined because there are no sevens in zero.
(b) 0 ÷ 7 is undefined because there are no sevens in zero.
(c) 7 ÷ 0 = 0 because if we split 7 cookies into 0 groups, each group will have 0 cookies.
(d) 0 ÷ 7 is undefined because we cannot split 0 cookies into 7 groups.
(e) 0 ÷ 7 = 0 because if we split 0 cookies into 7 groups, each group will have 0 cookies.
61. Our divisibility test for 3 in base ten is that 3 | n if and only if 3 divides the sum of the digits of n. When is a whole number written as a base six numeral divisible by 3?
(a) if and only if the sum of its digits is divisible by 3 (b) if and only if the sum of its digits is divisible by 5
(c) if and only if its last digit is 0 (d) if and only if its last digit is 3
(e) if and only if its last digit is 0 or 3
62. Suzanne asked the guy at the fish counter at Papa Joe’s for a pound of salmon. He selected a piece and put it on the digital scale. “It’s fifteen thousandths off,” he said.
Which one of these is a possible reading that the scale showed?
(a) 1.15 (b) 16.015 (c) 0.995 (d) 0.985 (e) 0.015
63. Which of the following is not a quotation from NCTM’s Principles and Standards?
(a) A curriculum is more than a collection of activities; it must be coherent, focused on important mathematics, and well articulated across the grades.
(b) From children’s earliest experiences with mathematics, it is important to help them understand that assertions should always have reasons.
(c) Instructional programs from kindergarten through grade 12 should enable stu- dents to build new mathematical knowledge through problem solving.
(d) The most important aspect of elementary-school mathematics education is teach- ing children to perform the arithmetical operations without using a calculator.
(e) Measurement is one of the most widely used applications of mathematics. It bridges two main areas of school mathematics—geometry and number.
64. What is the prime factorization of 10! ? (Recall that the exclamation point is the factorial operation; for example, 5! means 1 · 2 · 3 · 4 · 5.)
(a) 1 · 2 · 3 · 4 · 5 · 6 · 7 · 8 · 9 · 10 (b) 2 · 3 · 5 · 7
(c) 1 · 24· 32· 52· 7 (d) 1 · 28· 34· 52· 7
(e) 28· 34· 52· 7
65. Which one of the following arguments is valid?
(a) Premises: Every student who scored 30 or higher on the ACT mathematics test got a scholarship. Mary got a scholarship. Conclusion: Therefore Mary scored 30 or higher on the ACT mathematics test.
(b) Premises: Everyone who answered all the multiple choice questions on last term’s MTE 210 final exam correctly got at least a 3.0 grade in the course. John got a 2.8 course grade in MTE 210 last term. Conclusion: Therefore John answered all the multiple choice questions on last term’s final exam incorrectly.
(c) Premises: Everyone who answered all the multiple choice questions on last term’s MTE 210 final exam correctly got at least a 3.0 grade in the course. John did not answer all the multiple choice questions on last term’s final exam correctly.
Conclusion: Therefore John got less than a 3.0 course grade.
(d) Premises: If you don’t have enough income tax withheld from your paycheck, then you have to pay a penalty to the IRS. You didn’t have to pay a penalty to the IRS. Conclusion: Therefore you had enough income tax withheld from your paycheck.
(e) Premises: All Republicans support spending cuts. Some Republicans support tax increases. Conclusion: Therefore everyone who supports tax increases and spending cuts is a Republican.
66. Which one of these is not a reason that we (and humankind) expanded the number system beyond the whole numbers, the integers, the rational numbers, or the real numbers?
(a) to insure closure for arithmetic operations
(b) to be able to solve equations that otherwise had no answers
(c) to be able to apply mathematics to real-life situations in which the numbers we already had are insufficient
(d) out of a natural intellectual curiosity to increase our understanding of numbers (e) to be able to distinguish between average students and those with particular
mathematical talent
67. You and your best friend went on a week-long vacation together and agreed to split the costs down the middle. Over the week, each of you paid for various joint expenses (like gasoline, meal checks, motel bills), and kept track of what you had paid. At the end of the vacation it turned out that you had paid A dollars and your friend had paid B dollars. Assume that A < B. How much money must you give to your friend at this point to even things up?
(a) A + B
2 (b) B − A
2 (c) A − B
2 (d) B − A
(e) nothing—she owes you money
68. Suppose we want a nice, simple, easy-to-use fraction to approximate 193
581. Which of the following statements is the best description of what should be done?
(a) This fraction can be rounded to 190
580, so the approximation to use is 19 58. (b) This fraction can be rounded to 100
500, so the approximation to use is 1 5. (c) This fraction can be rounded to 200
600, so the approximation to use is 1
3, but we can’t immediately tell whether this approximation is actually a little larger or a little smaller than the original number.
(d) This fraction can be rounded to 200
600, so the approximation to use is 1 3, but this approximation is actually a little smaller than the original number, since we rounded both numerator and denominator up.
(e) This fraction can be rounded to 200
600 so the approximation to use is 1 3, but this approximation is actually a little larger than the original number, since we rounded both numerator and denominator up.
69. For which of these is a correct negation of the original statement provided?
(a) Statement: “Pamela is under six years old, and her father is supervising her play.”
Negation: “Pamela is not under six years old, and her father is not supervising her play.”
(b) Statement: “Pamela is under six years old, and her father is supervising her play.”
Negation: “Pamela is not under six years old, or her father is not supervising her play.”
(c) Statement: “If Pamela is under six years old, then her father is supervising her play.” Negation: “If Pamela is under six years old, then her father is not supervising her play.”
(d) Statement: “Some math teachers are members of the NCTM. Negation: “Some math teachers are not members of the NCTM.”
(e) Statement: “Some math teachers are members of the NCTM.” Negation: “All math teachers are members of the NCTM.”
70. What is the value of 72015+ 72014 72015− 72014? (a) −1 (b) 1 (c) 4
3 (d) 72014 (e) 74028
If you notice any mistakes on these pages, please let me know. As usual, you get one point extra credit if you are the first person to point out an error.
Jerrold W. Grossman Department of Mathematics and Statistics Oakland University November 20, 2015
