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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 38 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. 4