The Great American Scholar (GAS) Grants cover 100% of students’ tuition but no other expenses. In 1995, 4,000 GAS Grants were awarded, of which between 1⁄4 and 1⁄3 were mandated to go to public university students. In 1995, 50% of public university students’ costs went to tuition, while 85% of private university students’ costs went to tuition. Which of the following are possible total dollar values of all GAS Grants awarded in 1995 ?
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Solution
D
The grants cover just tuition, so you first need to find the costs of tuition for public and private university students in 1995. From the graph, a public student’s average total costs were about $6,000 and the question says that 50% of that went to tuition, or $3,000. For private students, average total costs were about $24,500 and 85% was went to tuition, or $20,825. Now you need to multiply those average tuition costs by the number of students in each university type that got a grant to get the total dollars awarded.Between 1⁄4 and 1⁄3 of the 4,000 awardees were public university students, or between 1,000 to 1,333 students.The other 2,667 to 3,000 were private university students. Since private tuition is more than public tuition, the high end of the total grant range will be if the most private students and the least public students got the grants: 3,000 private students × $20,825/student + 1,000 public students × $3,000/student = $65.48 million.The low end of the range will be if the most public school students and the least private students possible got the grants: 2,667 private students × $20,825/student + 1,333 public students × $3,000/student = $59.55 million.The only answer choice that falls between $59.55 and $65.48 million is choice (D).
2002 AIRPLANE INVENTORY FOR AIRLINES A AND B BY YEAR OF PURCHASE (as a percent of the 2002 inventory)
New regulations go into effect in 2003 that require all planes in inventory to be newer than ten years old. Each year following 2002, both airlines need to sell the planes the regulations force them to eliminate from inventory, and then use the proceeds of those sales to increase their inventory by 10% (rounded down because they are unable to buy fractions of planes). What is the combined number of planes owned by the two companies following their sales and purchases in 2004 ?
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Solution
626
You have to calculate how many planes each company sells and buys for 2003 and 2004 in order to calculate the total number of planes in 2004. In 2003, each company sells the planes they bought 10 years prior in 1993. For A: 250 × 10% = 25 planes removed; for B, 450 × 20% = 90 planes removed. In 2003 A has only 225, so they add 10%, or 22 new planes, for a new total of 247, and B has 360 left, so they add 36 for a new total of 396. In 2004, they sell the old planes they bought in 1994: For A, 250 × 4% = 10 planes; for B, 450 × 14% = 63 planes. Now A has 247 – 10 = 237, so they add 23 for a 2004 total of 260.B now has 396 – 63 = 333, so they add 33 for a 2004 total of 366. 260 + 366 = 626 planes in 2004.
Total crime incidents reported in Fairfax decreased by 25% from 2005 to 2010. For which of the crime categories presented in the graph was the percent change from 2005 to 2010 greater than the percent change of all crimes reported?
Indicate all possible values.
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Solution
A,C, and D
To find the percent change, use the percent change formula,\(\frac{difference}{Original}\) × 100, and be sure to use the values for 2005 as the original. Assaults changed by 40%, burglary changed by 62%, and auto theft changed by 30%, so choices (A), (C), and (D) are correct. Only domestic violence, which changed 21%, didn’t change more than 25%.
The number of assaults reported in Fairfax dropped every year from 2004 to 2010. For which year(s) was the rate of decrease greater than it had been the previous year?
Indicate all possible values.
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Solution
B and D
Rather than calculating, think in terms of the slope of the line connecting successive data points for assault. A greater rate of increase means a more sharply sloping downward line:The line from 2006 to 2007 is steeper than the one from 2005 to 2006, so choice (B) is correct, and the line from 2008 to 2009 is steeper than the one from 2007 to 2008, so choice (D) is also correct. For all the other years, the decrease in assaults is less than the previous year, so only choices (B) and (D) are correct.
Technology books have become an increasingly important subcategory of science/nature books. If technology books represent 30% of new adult science/nature books, half of used adult science/nature books, and \(\frac{1}{10}\) of children’s science nature books, how many technology books were sold in 2005?
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Solution
109
Use the second table in the diagram. First, to find the number of technology books in each category, you need to multiply the percentages or fractions by the total numbers of books sold for each respective category of science/nature books.Then add all those together to find the correct answer. For new adult: 30% × 150 = 50. For used adult: 0.5 × 70 = 35. For new children’s: 0.1 × 120 = 12. For used children’s: 0.1 × 120 = 12.The final answer is 50 + 35 + 12 + 12 = 109.
The following graph is a training log for a triathlete. It documents the number of hours she trained each week at each of three disciplines over a four week period.
In week 5, the athlete plans to decrease her training time in each sport by 10% to 20% of the hours she trained in week 4. Which of the following are possible numbers of hours she could bike in week 5?
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Solution
C
You can determine what hours make a 10 – 20% decrease by calculating 10% and 20% of the original value and subtracting, or, to save time, just calculate 90% and 80% of the original value. She biked 8.5 hours in week 4. 80% × 8.5 = 6.8, and 90% × 8.5 = 7.7, so any number of hours between 6.8 and 7.7 will be acceptable.Choice (C) is the only answer that falls in that range.