MATHEMATICS 320: OLD FINAL EXAM SOLUTIONS FROM FALL 2003

[Pages:3]MATHEMATICS 320: OLD FINAL EXAM SOLUTIONS FROM FALL 2003

INSTRUCTOR: MICHAEL B. SCOTT

(1) Write out 499 ? 3434 ? 77 as a single number to a single exponent. In your answer the base must be

in the simplest possible form. For example 2 ? 2 ? 2 ? 2 must be written as 24, not 42. (11 points)

Solution. 499 ? 3434 ? 77 = (72)9 ? (73)4 ? 77 = 718 ? 712 ? 77 = 737.

(2) The 9th term of an arithmetic sequence is 59 and the 13th term is 87. What is the initial term of

this sequence?

(11 points)

Solution. There are two unknowns in this problem, the initial term a and the common difference

d. We can use the following equations to solve for these unknowns, that is,

a + (13 - 1)d = 87 - a + (9 - 1)d = 59 ,

4d = 28

which means d = 7. Substituting d = 7 into the equation a + 8d = 59 and solving gives a = 3.

(3) If the side of a cube increases in size by 5%, then by what percentage does the volume increase?

Give your answer to the nearest one percent.

(11 points)

Solution. If s - 1.05s, then V = s3 - (1.05s)3 = 1.157625s3, which means the volume increase

16%.

(4) Find a subset of whole numbers that is closed for multiplication but not closed for addition. (11

points )

Solution. There are many answers. One would be the set of odd numbers, since the product of two

odd numbers is odd but the sum of any two odd numbers is even.

(5) Use the problem 43 - 27 to explain some of the advantages and disadvantages of using the standard

subtraction algorithm versus the subtract-from-the-base algorithm.

(11 points)

Solution.

We've computed 43 - 27 below using the standard algorithm on the left and the subtract-from-

the-base algorithm on the right.

43 13 -2 7

16

10 43 3 -2 7

16

The regrouping is more explicit in the subtract-from-the-base algorithm, but doing the actual subtraction, after regrouping, may be easier using the standard algorithm. This is not the only correct answer to this question. (6) Use the figure below to show that any convex 7-gon has the sum of its vertex angles equal to 900. (11 points)

Solution. Choosing any one vertex of the polygon we draw lines from the chosen vertex to the other

vertices as shown in the figure. This mean we can cover the 7-gon with 5 triangles so that the sum of the interior angles of all the triangles combined, each of which is 180, gives the sum of the vertex angles of the 7-gon. Thus, 5 ? 180 = 900.

Date: Spring 2004.

1

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INSTRUCTOR: MICHAEL B. SCOTT

(7) (a) Complete the following multiplication table in base five. ?1 2 3 4 112 3 4 2 2 4 11 13 3 3 11 14 22 4 4 13 22 31

(b) Compute 343five ? 12five. You can use your table from part (a). Solution.

3 4 3five

?

1 2five

1 2 4 1five

3 4 3 0five

1 0 2 2 1five

(7 points) (7 points)

(8) Determine whether 97 is a prime and explain how you would be sure.

(11 points)

Solution. By the Prime Factor Test, if 97 has any prime factors less than 97, then 97 must have at least one prime factor less than or equal to 97 10. So, we only have to check if the prime

numbers 2, 3, 5, or 7 divide 97. Since 2, 3, 5, and 7 do not divide 97, we can be sure that 97 is prime.

(9) If a = 2 ? 32 ? 53, find four possible values of b so that GCF(a, b) = 2 ? 52.

(11 points)

Solution. There are many answers for this problem. For the following values of b, we would have

GCF(a, b) = 2 ? 52. Let b = 2 ? 52 ? 7, b = 2 ? 52 ? 13, b = 2 ? 52 ? 17, and b = 2 ? 52 ? 19. The following

four values of b would also work: b = 2 ? 52, b = 22 ? 52, b = 23 ? 52, and b = 24 ? 52.

(10) State and Prove the divisibility test for 5 for a three digit number.

(11 points)

Hint: Start by letting abc represent a three-digit number and then expand by powers of ten.

Solution. Let abc represent a three-digit number where c = 0 or c = 5. Then

abc = 100a + 10b + 0 = 5(20a + 2b) or abc = 100a + 10b + 5 = 5(20a + 2b + 1).

In either case, 5 is a factor of the number abc, which means abc is divisible by 5.

(11)

Calculate

1.8 ? 1025 3.6 ? 1018

and

express

your

final

answer

in

scientific

notation.

Solution.

1.8 ? 1025 3.6 ? 1018

=

1.8 3.6

?

1025 1018

=

1 2

? 107

=

0.5 ? 107

=

5 ? 106.

(11 points)

(12) Use the following Venn diagram to represent and find the GCF and LCM of 42 and 60. Make sure

to label your diagram.

(11 points)

Solution. Since 42 = 2 ? 3 ? 7 and 60 = 22 ? 3 ? 5, we can represent the GCF as the intersection of the

set of factors of 42 and 60 and the LCM as the union of the set of factors of 42 and 60.

42 7

60

2

2

3

5

MATHEMATICS 320: OLD FINAL EXAM SOLUTIONS

(13) Let m(1) = m(2), prove that l m. Hint: Assume that if corresponding angles are equal, then l m.

FROM FALL 2003

3

(11 points)

l 1

2 m

3

Solution. Since 2 and 3 are vertical angles, m(2) = m(3). We are given that m(1) = m(2), so m(1) = m(3) by substitution. Therefore, because 1 and 3 are corresponding and congruent, we can conclude l m. (14) Inga was making a cake that called for 4 cups of flour. However, she could only find a two-thirds measuring cup. How many two-thirds measuring cups of flour will she need to make her cake? (11 points) Solution. We need to know how 2/3s are in 4, which is a division problem. Thus, Inga needs

4

?

2 3

=

4

?

3 2

=

12 2

=

6

two-thirds measuring cups to make 4 cups.

(15) A television set to be sold at a 13% discount, which amounted to $78. How much would the set sell

for after the discount?

(11 points)

Solution. We can solve this problem using a proportion as follows:

part whole

=

13 100

=

78 original

price .

Solving, the original price of the television set is

100 ? 78 13

=

$600.

Therefore, the television would sell for $522 after the discount.

(16) Can you use lattice multiplication for decimals? For example, how would you multiply 3.34 times

1.7?

(11 points)

Solution.

?3 3 4

000 3 3 4

1

5

222 1 1 8

7

6 7 8

We conclude 3.34 ? 1.7 = 5.678. Notice that the lattice method keeps track of the decimal point

even though the decimal point wasn't explicitly written.

(17) A student in your class says that if the ratio of oil to vinegar in a salad dressing is 3:4, that means

that 75% of the salad dressing is oil. Another student says the dressing is less than 50% oil. Explain

the correct answer.

(11 points)

Solution. In this problem, the ratio 3:4 is a part-to-part ratio, and the students are making

inferences from the part-to-whole ratio of oil to salad dressing, which is 3 : (3 + 4) = 3 : 7. This

means the student who said the dressing is less than 50% oil is correct.

Department of Mathematics, Kansas State University E-mail address: mbscott@math.ksu.edu

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