Math 1030-7 Midterm 1
Solution
Instructions: There are TWO parts in this exam, FIVE problems in each
part. Answer the questions in the spaces provided on the question sheets.
If you need more space, use the bottom of the last page, but remember to
indicate which problem you are doing. Partial credit will be awarded. Scientific
calculators are allowed. This exam is closed book and closed notes. Show you
work on each problem.
Part I: Basic problems
1. (10 points) Draw a Venn diagram for the following proposition and label all regions in your diagram.
All widows are women.
Solution:
People
Women
Widows
2. (10 points) Convert 5000 meters into miles. (1 mi = 1609.3 m)
Solution: First, we can derive the conversion formula
1=
1609.3 m
1 mi
=
1609.3 m
1 mi
Then we have
5000 m = 5000 m × 1 = 5000 m ×
5000
1 mi
=
mi ≈ 3.1 mi
1609.3 m
1609.3
3. (10 points) Convert 110 square yards into square feet.
Solution: We know
1 yd = 3 ft
Take square on both side
(1 yd)2 = (3 ft)2
1 yd2 = 9 ft2
1=
1 yd2
9 ft2
2 =
9 ft
1 yd2
Then we have
110 yd2 = 110 yd2 × 1 = 110 yd2 ×
1
9 ft2
= 110 × 9 ft2 = 990 ft2
1 yd2
4. (10 points) The population of the United States increased from 249 million in 1990 to 308 million
in 2010. Find the absolute change and relative change of the population.
Solution: By the definition of absolute change and relative change,
absolute change = new value−old value = 308 million−249 million = 59 million
relative change =
59 million
absolute change
=
≈ 0.2369 = 23.69%
old value
249 million
5. (10 points) Use the following table to answer the question. What percentage of all secondary
teachers are female?
Gender
Men
Women
Elementary and middle school teachers
562,113
2,380,887
Secondary teachers
499,098
658,902
Solution: This percentage is used as fraction by the keyword “of”.
fraction =
female secondary teachers
658902
=
= 0.569 = 56.9%
all secondary teachers
499098 + 658902
Part II: Advanced problems
6. Consider the following deductive argument:
Premise:
All great basketball players must wear expensive shoes.
Premise:
Shaquille O’Neal is a great basketball player.
Conclusion: He must wear expensive shoes.
(a) (5 points) Draw a Venn diagram for the first premise and label all regions in your diagram.
(b) (5 points) Check the validity of the argument using your diagram.
Solution: Valid.
People
People who
wear expensive shoes
Great
basketball
players
b Shaquille
O’neal
7. (10 points) You plan to travel to Japan and you are practicing how to deal with different measurement systems. Consider the following situation. Suppose pears at a store in Japan are priced at
950 yen per kilogram. What is the price of these pears in US dollars per pound if 1 US dollar is
117.9 yen and 1 kilogram is 2.2 pounds.
2
Solution: We have two conversion formulaes:
1=
117.9 yen
1$
=
117.9 yen
1$
1=
1 kg
2.2 lb
=
2.2 lb
1 kg
Then,
950 yen/kg =
950 yen
1$
1 kg
950
950 yen
×1 =
×
×
=
$/lb ≈ 3.66 $/lb
1 kg
1 kg
117.9 yen 2.2 lb
117.9 × 2.2
8. Suppose your car averages 38 miles per gallon of gasoline on the highway if your average speed is
55 miles per hour.
(a) (5 points) What is the driving time for a 2000-mile trip if you drive at an average speed of 55
miles per hour?
(b) (5 points) Assume a gasoline price of $2.90 per gallon. What is the gasoline cost for a 2000-mile
trip if you drive at an average speed of 55 miles per hour?
Solution:
(a) The question is asking for driving time, so the unit of the answer is expected to be hour.
2000 mi
2000
=
hr ≈ 36.36 hr
55 mi/hr
55
(b) The question is asking for gasoline cost, so the unit of the answer is expected to be US dollar.
2.90 $/gal ×
1
2.9 × 2000
× 2000 mi =
$ ≈ 152.63 $
38 mi/gal
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9. London (Reuters) 11/15/04:“U.S. light crude fell $1.80 to $45.60 a barrel, the lowest price since
mid-September.”
(a) (3 points) Find the original price of U.S. light crude.
(b) (7 points) By what percentage did the price change?
Solution:
(a) 45.60 $ + 1.80 $ = 47.4 $
(b) By the definition of relative change
relative change =
absolute change
−1.80 $
=
≈ −0.038 = −3.8%
old value
47.4 $
10. (10 points) If a 34 ounce can of olives is 183% more money than a 12 ounce can of olives, which is
a better deal? Explain.
3
Solution: Suppose the price of a 12 ounce can of olives is $1. Then the price of a 34
ounce can of olives is
1 $ × (1 + 183%) = 2.83 $
The unit price of a 12 ounce can is
1 $ ÷ 12 ounce ≈ 0.0833 $/ounce
The unit price of a 34 ounce can is
2.83 $ ÷ 34 ounce ≈ 0.0832 $/ounce
So the 34 ounce can of olives is a better deal.
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