Math 105 Test 1 Answer Key

MATH 105, Fall 2008 Instructor: Hao-Nhien Q. Vu

_________ANSWER KEY________

EXAM NO. 1

Page 1

SHOW YOUR WORK. Put your answer in the column on the right. You have 60 minutes to complete the exam. 1. (10 points) Determine the type of reasoning that was used, and circle Inductive or Deductive on the right. a. “My mom always cuts off the ends of the ham before cooking it, and it always comes out tasting superb. So I cut off the ends of the ham as well.”

a. Inductive Deductive

b. “The 10% richest people will always own much more than 10% of our country’s wealth. That’s what it means to be the richest. If they were holding 10%, they’d be just mathematically average.”

b. Inductive

Deductive

c. If x < 5 then –x > –5.

c. Inductive

Deductive

d. “Don’t worry about it, nobody has died playing Quidditch before.”

d. Inductive Deductive

e. “Throughout history, men leaders have done nothing but started wars. Women leaders will bring peace.”

e. Inductive Deductive I accept both answers. The first statement is inductive; the second is (incorrect) deductive.

2. (2 points) Fill in the blank: Two __counterexamples____ to the statement in 1.e are Queen Victoria and Prime Minister Margaret Thatcher. Queen Victoria’s England fought the Opium War in China, the Crimean War in Europe and the Boer Wars in South Africa. Prime Minister Thatcher engaged in the Falklands/Malvinas War. 3. (8 points) Use inductive reasoning to determine the next three numbers in the pattern. Write your answer in the space to the right. a. 1, 13, 25, 37, 49, …

a. _61__, __73__, __85__

b. –1, 2, –4, 8, –16, …

b. _32__, _–64 __, _128_

4. (4 points) In the picture below, if each square represents one square-foot, estimate the area of the shaded figure. Write your answer to the right. You get: 4 points for any estimate between 15 to 25. 2 points if down to 10 or up to 30. 1 point for up to 48.

4. ___________________

MATH 105, Fall 2008

_________ANSWER KEY________

Instructor: Hao-Nhien Q. Vu

EXAM NO. 1

Page 2

5. Solve and place your answers in the spaces to the right. a. (4 points) What percent of 500 is 2.5? (Percent) (Base) = (Amount) x 500 = 2.5 ==>

a. ___0.5%__________ x = 0.005 or 0.5%

b. (4 points) What is 3% of 25?

b. ___.75_________

(Percent) (Base) = (Amount) (3%) (25) = (0.03) (25) = .75 c. (4 points) 25% of what is 500?

c. ____2000__________

(Percent) (Base) = (Amount) (25%) x = 500 ==> x = 2000 d. (4 points) How much is 15% tip on $30?

d. ____$4.50________

(Percent) (Base) = (Amount) (15%) ($30) = (0.15) ($30) = $4.50 6. Solve and place your answers in the spaces to the right. For all the subparts of this problem, use the following sets: A = {a, c, e, g} B = {a, b, c, f, g} C = {a, e, f, g} a. (4 points) Write the roster form of A ∩ B

a. ___{a, c, g}_____

b. (4 points) Write the roster form of A ∪ B

b. ___{a, b, c, e, f, g}_______

c. (4 points) Write the roster form of A ∪ C

c. ___{a, c, e, f, g}_____

d. (4 points) Write the roster form of B ∩ C

a. ___{a, f, g}______

MATH 105, Fall 2008

_________ANSWER KEY________

Instructor: Hao-Nhien Q. Vu

EXAM NO. 1

Page 3

7. (14 points) True or false (circle the answer on the right). a. {7} ⊆ {x | x ∈ N and x ≤ 7 } The set on the right is: {0, 1, 2, 3, 4, 5, 6, 7} So, yes, the set {7} is a subset of it.

a. True

False

b. {c, o, w} has eight subsets. A set of n elements has 2n subsets. This set has 3 elements, so it has 23 subsets = 8.

b. True

False

c. {a | a ∈ N and a ≤ 10 } is infinite. This set is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Not infinite.

c. True

False

d. These two sets are the same: y 1 {(x, y) | −x= } 2 2

d. True

False

e. True

False

f. True

False

f. True

False

and

{(x, y) | y = 2x + 1} The two equations are the same, so the (x, y) satisfying them are the same.

e. For any set A,

∅⊆ A

The empty set is a subset of all sets. f. For any set A, A ∪

∅=A

Combining the “elements” of an empty set with A doesn’t change A. g. {€, £, ¥} = {¢, ¥, £} They don’t have the same elements.

8. (6 points) Convert the following and write your answer in the space to the right. a. 3,000 meters to kilometers:

______3________ km

b. 0.5 grams to milligrams:

______500_______ mg

c. 2 liters to milliliters:

______2000______ ml

MATH 105, Fall 2008

_________ANSWER KEY________

Instructor: Hao-Nhien Q. Vu

EXAM NO. 1

Page 4

9. (4 points) Convert to kilometer and find the number in the blank. Fill in the blank. 400m + 42.195km + 100m + ____7.305_____km = 50km Let x be the amount in the blank. Convert everything to kilometers: 0.4 + 42.195 + 0.1 + x = 50 ==> 42.695 + x = 50 ==> x = 50 – 42.695 = 7.305 10. a. (3 points) Fill in the next line in Pascal’s triangle below: 1 1 1 1 1 ____1

3 4

5 6

1 2

1 3

6 10

15

1 4

10 20

1 5

15

1 6

1____

b. (3 points) Use Pascal’s triangle to fill in the blanks below: (a + b) 5 = a 5 + 5 a 4b + 10 a 3b 2 + 10 a 2b 3 + 5

ab 4 + b 5

11. (2 points) Write 2013 in Roman numerals.

11. ___MMXIII_____

12. Suppose there is a positional (i.e. place-value) numeration system that only has two symbols: ► means 1

and

♥ means 5.

Like our system, it uses base 10. Like the Babylonian system, it combines these two symbols to get various values between 1 and 9. a. (3 points) Write 7 using this numeration system.

♥►► a. ________________

b. (3 points) Write 22 using this numeration system. Remember: This is supposed to be a positional system, so the numerals have place value (units, tens, hundreds, etc.)

►► ►► b. ________________

MATH 105, Fall 2008 Instructor: Hao-Nhien Q. Vu

_________ANSWER KEY________

EXAM NO. 1

Page 5

13. (6 points) In your own words, describe one advantage of using a positional (i.e. place-value) numeration system. Describe the importance of a symbol for zero in a positional numeration system; give at least one concrete example.

Your answer needs to include both items asked for in the question: (1) One advantage of a positional numeration system; and (2) The importance of a symbol for zero. The positional numeration system allows for ease of calculations. Adding, subtracting, multiplying, dividing become easier to learn and practical to use. The numbers be to used directly for calculation without requiring an abacus or any other outside calculation tools. The symbol for zero is necessary to avoid confusion. It is a placeholder that allows us to distinguish numerals in one position (or place-value) from another. A positional system that doesn’t have a symbol for zero can be confusing. For example, the Babylonians did not have a symbol for zero, and the Babylonian numerals that we examined in class can be either 10 or 600.