Mathematics Test with Rationales

What unit of measurement could be used to describe the spread of a forest fire in a unit time? A. 10 square yards per second B. 10 yards per minute C. 10 feet per hour D. 10 cubic feet per hour

A. 10 square yards per second The only appropriate answer is the one that describes an area of forest consumed per unit time. Only answer A is a unit of area measurement.

A drawer contains 5 black socks, 3 blue socks, and 2 red socks. Without looking at the socks you draw, what is the probability that you will draw two black socks in a row? A. 2/9 B. 1/4 C. 17/18 D. 1/18

A. 2/9 In this example of conditional probability, the probability of drawing a black sock on the first draw is 5/10. The problem implies that there is no replacement, so the probability of drawing a black sock on the second draw is 4/9. Multiply the two probabilities and reduce to lowest terms.

Jesse is trying to find out how many boys attend his school. To get an estimate, he counts the girls and boys in 3 of his classes and finds there are 58 boys and 62 girls. If his school has a total of 560 students how many boys attend his school? A. 271 B. 289 C. 325 D. 347

A. 271 Find the proportion of boys in the classes he counted. Use this to find the number in the entire school by setting up a proportion. (58 boys)/(58 + 62 total students) = x boys/560 --> 270.7 --> 271 boys

What is the probability of flipping 3 coins and getting 2 tails and 1 head? A. 3/8 B. 1/8 C. 1/3 D. 1/2

A. 3/8 The simplest way to find the answer can be found by listing out all the possible combinations or making a tree diagram. There are 8 total combinations. Three result in 2 tails and 1 head.

Which of the following is true about a parallelogram? A. The opposite sides are congruent. B. The diagonals are congruent. C. All four angles are congruent. D. The sum of the interior angles is 180 degrees.

A. Opposite sides are congruent Choices B and C are true for rectangles. Choice D is true for triangles.

What is the equation of the graph shown below? 5| 4| 3| 2|o 1| o -5 -4 -3 -2 -1 0 1 2 3 4 5 -1| -2| -3| -4| -5| A. x + 2y = 4 B. x - 2y = 4 C. 2x + y = 4 D. x + 2y = -4

A. x + 2y = 4 When y = 0, solving for x yields 4, giving the point (4, 0). When x = 0, solving y yields 2, giving the point (0, 2). These are the intercepts of the given graph.

Which of the following equations has the same slope as 6x + 3y = 9? A. y = -2x + 8 B. y = 2x + 8 C. y = 6x + 8 D. y = -6x + 8

A. y = -2x + 8 Put 6x + 3y = 9 into slope-intercept form, y = mx + b. y = -2x + 3 m, slope, is -2, so the equation with an x coefficient of -2 has the same slope.

The price of gas is $3.27 per gallon. Your tank holds 15 gallons of fuel. You are using two tanks of gas a week. Approximately how much will you save each week if the price of gas goes down to $2.30 per gallon? A. $26.00 B. $29.00 C. $15.00 D. $17.00

B. $29.00 15 gallons * 2 tanks = 30 gallons a week = 30 gallons * $3.27 = $98.10 30 gallons * $2.30 = $69.00 $98.10 - $69.00 = $29.10 is approximately $29.00.

The fees charged by a parking garage are as follows: hours 1 2 3 4 5 fee $12 $19 $26 $33 $40 How would you summarize the fees charged? A. $12 an hour B. $5 plus $7 per hour C. $15 an hour with a $3 discount D. $4 plus $8 per hour

B. $5 plus $7 per hour Choice B satisfies each entry in the table. For instance, at 4 hours the calculation is 5 + 7(4) = 33

At the lake, the ratio is 16 ducks to 9 geese. Suppose there are 192 ducks. How many geese are there? A. 21 B. 108 C. 7 D. 1,728

B. 108 Set up the proportion: 16/9 = 192/g. Solve for g. 16g = (192)(9) 16g = 1728 g = 108

Solve for x: 7 - 5x = 7x - 11 - 3x A. -2 B. 2 C. 0 D. -1 1/2

B. 2 First add 5x to both sides to get all the x terms on one side. Then 7 = 7x - 11 - 3x + 5x. Combine all the x terms: 7 = 9x - 11. Add 11 to both sides to get 18 = 9x. Dividing both sides by 9, x = 2.

Choose the expression that is NOT equivalent to 5x + 3y + 15z A. 5(x + 3z) + 3y B. 3(x + y + 5z) C. 3y + 5(x + 3z) D. 5x + 3(y + 5z)

B. 3(x + y + 5z) 5x + 3y + 15z = (5x + 15z) + 3y = 5(x + 3z) + 3y A is equivalent = 5x + (3y + 15z) = 5x + 3(y + 5z) D is equivalent = 3y + (5x + 15z) = 3y + 5(x + 3z) C is equivalent We can solve all of these using the associative property and then factoring. However, in B 3(x + y + 5z) by distributive property = 3x + 3y + 15z, which does not equal 5x + 3y + 15z.

Which choice below represents the phrase "3 less than 5 times a number?" A. 3 - 5n B. 5n - 3 C. 3 < 5n D. 5(n - 3)

B. 5n - 3 Choice B expresses the operations in the correct order. Choice C expresses 3 is less than 5 times a number.

The following chart shows the yearly average number of international tourists visiting Palm Beach for 1990 - 1994. How many more international tourists visited Palm Beach in 1994 than in 1991? 2,000,000 |---------------------------- | | | 1,500,000 |---------------|------------| | | | | | | | 1,000,000 |---------|-----|------|-----| | | | | | | | | | | | 500,000 |---|-----|-----|------|------| | | | | | | | | | | | | 0 | 1990 1991 1992 1993 1994 A. 100,000 B. 600,000 C. 1,600,000 D. 8,000,000

B. 600,000 The number of tourists in 1991 was 1,000,000 and the number in 1994 was 1,600,000. Subtract to get the difference of 600,000.

Which number line shows the solution to 7x - 5 >= 9x - 17? A. <----------------------------> -6 0 6 *------> B. <----------------------------> <-- -6 ---- 0 ----* 6 C. <----------------------------> -6 0 6 o-----> D. <----------------------------> -6*-----0----*6

B. <--<----(-6)-----0-----(6)----> First gather all the x-terms on one side of the inequality and the numbers on the other. 7x - 5 >= 9x - 17 -2x >= -12 Dividing both sides of an inequality by a negative number reverses the inequality sign. So division by -2 on both sides results in x <= 6, which is graphed in choice B.

Corporate salaries are listed for several employees. Which is the best measure of central tendency? $24,000, $24,000, $26,000, $28,000, $30,000, $120,000 A. mean B. median C. mode D. there is no best measure

B. Median The median provides the best measure of central tendency in this case because the mode is the lowest number and the mean is disproportionately skewed by the outlier $120,000.

Mary did comparison shopping on her favorite brand of coffee. Over half of the stores priced the coffee at $1.70. Most of the remaining stores priced the coffee at $1.80, except for a few who charged $1.90. Which of the following statements about the distribution of prices is true? A. The mean and the mode are the same B. The mean is greater than the mode C. The mean is less than the mode D. The mean is less than the median

B. The mean is greater than the mode Over half the stores priced the coffee at $1.70, so this is the mode. The mean is slightly over $1.70 because other stores priced the coffee at over $1.70.

If n represents a number, which one of the following is the correct interpretation for the expression 8 - 2n? A. Six times a number B. Twice a number subtracted from eight C. Twice eight minus a number D. Eight subtracted from twice a number

B. Twice a number subtracted from eight The expression 2n means twice a number. Since 8 precedes 2n in the given expression, this means that 2n is being subtracted from 8.

(-2.1 X 10^4)(4.2 X 10^-5) = A. 8.82 B. -8.82 C. -0.882 D. 0.882

C. -0.882 First, multiply -2.1 by 4.2 to get -8.82. Then, multiply 10^4 by 10^-5 to get 10^-1. -8.82 * 10^-1 = -0.882

Put the following in order from smallest to largest: 3^2, -pi, 2^3, 4 2/3, -5, -8.1, sqrt(6) A. -5, -8.1, sqrt(6), -pi, 4 2/3, 3^2, 2^3 B. -5, -8.1, - pi, sqrt(6), 4 2/3, 2^3, 3^2 C. -8.1, -5, - pi, sqrt(6), 4 2/3, 2^3, 3^2 D. 3^2, - pi, 2^3, 4 2/3, -5, -8.1, sqrt(6)

C. -8.1, -5, -pi, sqrt(6), 4 2/3, 2^3, 3^2 First put the numbers in decimal form so they may be easily compared, 3^2 = 9, -pi ~ -3.14, 2^3 = 8, 4 2/3 = 4.67, sqrt (6) ~ 2.45 9, -3.14, 8, 4.67, -5, -8.1, 2.45 Then arrange in order beginning with the largest negative number: -8.1, -5, -3.14, 2.45, 4.67, 8, 9 --> -8.1, -5, -pi, sqrt(6), 4 2/3, 2^3, 3^2

The table below shows the distribution of majors for a group of college students. Major Proportion of students Mathematics 0.32 Photography 0.26 Journalism 0.19 Engineering 0.21 Criminal Law 0.02 If we know that a student, chosen at random, is not majoring in mathematics or engineering, what is the probability that the student is majoring in journalism? A. 0.19 B. 0.36 C. 0.40 D. 0.81

C. 0.40 The proportion of students majoring in math or engineering is 0.32 + 0.21 = 0.53. This means that the proportion of students not majoring in math or engineering is 1.00 - 0.53 = 0.47. The proportion of students majoring in journalism out of those not majoring in math or engineering is 0.19/0.47 = 0.404.

The admission price to tour the Haunted House has been changed from $25 to $30. What is the percentage of change in the admission price? A. 5% B. 16.7% C. 20% D. 25%

C. 20% First use subtraction to find the change in dollars: 30 - 25 = 5. Then compare the change to the original to find the percent: 5/25 = 0.2 = 20%

How many unique sandwiches can be made considering choices of white or wheat bread, meat choices of turkey, ham or roast beef, and an option of butter, mayonnaise, mustard, or no condiment? A. 234 B. 64 C. 24 D. 9

C. 24 Model the Fundamental Counting Principle wit 3 events: 2 bread choices, 3 meat choices, and 4 possibilites for condiments. 2 * 3 * 4 = 24

If a cylinder has a radius of 5 and a height of 10 , find the volume. A. 50pi B. 100pi C. 250pi D. 500pi

C. 250 pi Applying the formula for the volume of a cylinder, V = pi r^2 h, gives pi * 5^2 * 10 = 250 pi

Choose the set in which the members are not equivalent. A. 1/2, 0.5, 50% B. 10/5, 2.0, 200% C. 3/8, 0.385, 38.5% D. 7/10, 0.7, 70%

C. 3/8, 0.385, 38.5% 3/8 is equivalent to .375 and 37.5%

Find the surface area of a box 3 feet wide, 5 feet tall, and 4 feet deep. A. 47 sq. ft. B. 60 sq. ft. C. 94 sq. ft. D. 188 sq. ft.

C. 94 sq. ft The top and bottom faces measure 3 ft. by 4 ft., so the surface area of teh top and bottom together is 2(12) = 24. The front and back measure 3 ft. by 5 ft., so the surface area of those two faces is 2(15) = 30. The left and right sides measure 4 ft. by 5 ft., so the surface area of those two faces is 2(20) = 40. The total surface area is therefore 24 + 30 + 40 = 94 square feet.

Which statement is true about George's budget? ------- Misc. ------- Food -------------- Utilities ---------------------------- Mortgage A. George spends the greatest portion of his income on food B. George spends twice as much on utilities as he does on his mortgage C. George spends twice as much on utilities as he does on food D. George spends the same amount on food and utilities as he does on mortgage

C. George spends twice as much on utilities as he does on food. The wedge representing utilities is twice as large as the wedge representing food.

A student has taken three tests in his algebra class for which the mean score is 88. He will take one more test and his final grade will be the mean of all four tests. He wants to achieve a final grade of 90. Which one of the following is the correct procedure to determine the score he needs on the fourth test? A. He needs a score of 92 since (88 + 92)/2 = 90 B. He needs a score of 89.5 since (88 +90 + 90 + 90)/4 = 90 C. He needs a score of 96 since (88 + 88 + 88 + 96)/4 = 90 D. He cannot achieve a final grade of 90 since each of his scores on the first three tests is less than 90

C. He needs a score of 96 since (88 + 88 + 88 + 96)/4 = 90 The sum of all four tests must be (90)(4) = 360 in order to achieve a mean score of 90. Since he has averaged 88 on his first three tests, the sum of his scores thus far is (88)(3) = 264. Therefore he needs a score of 360 - 264 = 96 on his fourth test.

Insert mathematical symbols to make the given calculation correct. 3 + 5 * 4 - 1 = 18 A. Place parentheses around 5 * 4 B. Place parentheses around 3 + 5 C. Place parentheses around 4 - 1 D. No symbols are needed. The calculation is already correct

C. Place parentheses around 4 - 1 Without inserting any symbols, the problem's answer is 22 as multiplication must be completed before the addition and subtraction. Choice C turns the calculation into 3 + 5 * (4 - 1) 3 + 5 * 3 3 + 15 18

In many states, you can get a driver's license when you turn 16. Which of the age statistic choices below would best represent when a person gets a license? A. mean B. median C. mode D. variance

C. mode People younger than 16 can't get a license, so the mean will be skewed by people older than 16 getting a license for the first time. The median will also be skewed in this same manner.

Which of the following sets of ordered pairs does not represent a function? A. {(1, -1), (2, -2), (3, -3)} B. {(3, 1), (4, 1), (5, 1)} C. {(1, 3), (1, 4), (1, 5)} D. All of the above do represent functions.

C. {(3, 1), (4, 1), (5, 1)} A function cannot have more than one output value (y) for the same input (x).

Which choice below could correctly appear during the solving of the given equation? 15 + 3x = -8x A. 15 = -5x B. 18 = -8x C. 15 = 11x D. 15 + 11x = 0

D. 15 + 11x = 0 Choice D arises when 8x is added to both sides of the equation.

In a card game, you get another turn if the card you draw is red or if it is a jack, queen, king, or ace. You are the first to draw from a full deck of 52 cards. What is the chance you will get another turn? A. 2/13 B. 4/13 C. 1/2 D. 17/26

D. 17/26 The cards that would give you another turn include all 26 red cards, plus two black jacks, queens, kings, and aces, making a total of 34 out of 52 or 17/26.

Simplify: (2 2/5)/(2/3) A. 2 2/5 B. 1 3/5 c. 2 3/5 D. 3 3/5

D. 3 3/5 First convert to improper fractions, (12/5) / (2/3) Change to a multiplication problem by using teh reciprocal of teh fraction after the division sign, cancel common factors, multiply numerators together and denominators together. Simplify (12/5) * (3/2) = (6/5) * (3/1) = 18/5 = 3 3/5

On a map, 10 miles corresponds to 3/4 inch. If two circles are 3 inches apart on this map, what is the actual distance in miles between them? A. 12.25 B. 22.5 C. 30 D. 40

D. 40 Let x represent the distance in miles between the two cities. Then (3/4)/3 = 10/x. Cross-multiply to get 3/4x = (10)(3) = 30. Then x = 30/(3/4) = 40.

Anna can run 5 miles per hour. How many feet per minute does she run? A. 0.06 feet/minute B. 17.6 feet/minute C. 63,360 feet/minute D. 440 feet/minute

D. 440 feet/minute (5 miles)/(1 hour) * (1 hour)/(60 minutes) * (5280 feet)/(1 mile) = 440 feet/minute

The figure below shows a running track in the shape of a rectangle with semicircles at each end. Calculate the distance around the track. C__________________________D ( | | ) B( | 3y | )E ( |__________________________| ) A 7x F A. 6(pi)y + 14x B. 3(pi)y + 7x C. 6(pi)y + 7x D. 3(pi)y + 14x

D. ? no multiple choice answers provided in book The two semicircles of the track create one circle with a diameter 3y. The circumference of a circle is C = pi d so C = 3 pi y. The length of both sides of the track is 7x each side, so the total circumference around the track is 3 pi y + 7x + 7x = 3 pi y + 14x

Hiroshi surveys his classmates to find what percent of them come to school on the bus, by car, by subway, by bicycle, or on foot. What is the best way to display his results? A. A line graph B. A box plot C. A scatterplot D. A circle graph

D. A circle graph A circle graph, or pie chart, typically represents percent values.

Solve |2x - 5| + 2 = 45 A. -19 B. 19 C. 24 D. Both A and C

D. Both A and C First isolate the absolute value expression, then set up two equations to solve. |2x - 5| = 43 2x - 5 = -43 or 2x - 5 = 43 2x = -38 or 2x = 48 x = -19 or x = 24

Find the zeros of the function f(x) = x^2 - 12x - 13 A. -1 B. 0 C. 13 D. Both A and C

D. Both A and C Set f(x) equal to 0 and solve by factoring: x^2 - 12x - 13 = 0 (x - 13)(x + 1) = 0 x = 13, -1

Which of the following statements is false about the data: 2, 4, 6, 8, 10? A. The mean is 6 B. The median is 6 C. There is no mode D. The range is 10

D. The range is 10 In this case that median is 6, which is also the mean: (2 + 4 + 6 + 8 + 10)/5 = 30/5 Since none of the data values appear more than once, there is no mode. However, the range is 10 - 2 = 8, not 10, so choice D is false.

The ages of the participants in a hula-hoop contest are as follows: 10, 18, 22, 17, 77, 19, 13, 20, 10, 15. Which measure would most accurately represent the data as a whole? A. range B. mode C. mean D. median

D. median The data set contains an outlier. The range including teh outlier would represent too wide a variation of the data, most of which falls within a narrow range. The mode would be a value smaller than all the other data items. The mean would be too large, skewed by the presence of the outlier. Only the median would have a value representative of the data as a whole.

The owner of a rectangular piece of land 40 yards in length and 30 yards in width wants to divide it into two parts. She plans to join two opposite corners with a fence as shown in the diagram below. The cost of the fence will be approximately $25 per linear foot. What is the estimated total cost for the fence? A. $1,250 B. $62,500 C. $5,250 D. $3,750

D. no multiple-choice answers provided in book Find the length of the diagonal by using the Pythagorean theorem. Let x be the length of the diagonal. 30^2 + 40^2 = x^2 --> 900 + 1600 = x^2 2500 = x^2 --> sqrt(2500) = sqrt(x^2) x = 50 yards Convert to feet. (50 yards)/x feet = (1 yard)/(3 feet) --> 150 feet The fence costs $25 per linear foot, so the cost is (150 ft)($25) = $3,750.

Solve for y: 4x - 3y = 9 A. y = 4x - 9 B. y = 4x + 3 C. y = 3 - (4/3)x D. y = (4/3)x - 3

D. y = (4/3)x - 3 Original equation: 4x - 3y = 9, subtract 4x from both sides -3y = -4x + 9, divide each term by -3 y = (4/3)x - 3

Which equation matches the data in the table? x 3 4 5 6 y 7 8 9 10 A. y = 2x - 1 B. y = 2x + 1 C. y = -x + 10 D. y = x + 4

D. y = x + 4 Each y value is 4 greater than its x value, which is the relationship demonstrated by choice D.