PCAT Quantitative ability

Your supervisor instructs you to purchase 240 pens and 6 staplers for the nurse's station. Pens are purchased in sets of 6 for $2.35 per pack. Staplers are sold in sets of 2 for 12.95. How much will purchasing these products cost? A. $132.85 B. $145.75 C. $162.90 D. $225.25 E. $226.75

A. $132.85 You will need 40 packs of pens and 3 sets of staplers. Thus, the total cost may be represented by the expression, 40(2.35) + 3(12.95). The total cost is $132.85.

Jim is able to sell a hand-carved statue for $670 which was a 35% profit over his cost. How much did the statue originally cost him? A. $496.30 B. $512.40 C. $555.40 D. $574.90 E. $588.20

A. $496.30 $670 = Cost + 0.35(Cost) = 1.35(Cost), Cost = $670/1.35 = $496.30

A straight line with slope +4 is plotted on a standard Cartesian (xy) coordinate system so that it intersects the y-axis at a value of y = 1. Which of the following points will the line pass through? A. (2,9) B. (0,-1) C. (0,0) D. (4,1) E. (1,4)

A. (2,9) As defined, the line can be described by the equation. Expression A fits this equation: . The others do not

5. Solve the following equation for A : 2A/3 = 8 + 4A A. -2.4 B. 2.4 C. 1.3 D. -1.3 E. 0

A. -2.4 In order to solve for A, both sides of the equation may first be multiplied by 3. This is written as 3(2A/3)=3(8+4A) or 2A=24+12A. Subtraction of 12A from both sides of the equation gives -10A=24. Division by -10 gives A = -2.4

A drawer contains only socks and gloves. There are twice as many socks as gloves. The socks are either red or yellow, and 4 times as many socks are red as are yellow. If one article is to be drawn at random from the drawer, what is the probability that the article of clothing drawn will be a yellow sock? A. 0.133 B. 0.2 C. 0.25 D. 0.5

A. 0.133 We want to determine the probability of drawing a certain type of sock. With probability, the key is to break it into its components and then put the proper ratios together. socks = 2 gloves socks: red = 4 yellow probability of drawing sock: 2/3 probability of drawing yellow sock: 1/5 probability of yellow(1/5) sock(2/3): 1/5 x 2/3 = 2/15 = .133

If 300 jellybeans cost you x dollars. How many jellybeans can you purchase for 50 cents at the same rate? A. 150/x B. 150x C. 6x D. 1500/x E. 600x

A. 150/x 50 cents is half of one dollar, thus the ratio is written as half of 300, or 150, to x. The equation representing this situation is 300/x*1/2=150/x.

2. If Sally can paint a house in 4 hours, and John can paint the same house in 6 hour, how long will it take for both of them to paint the house together? A. 2 hours and 24 minutes B. 3 hours and 12 minutes C. 3 hours and 44 minutes D. 4 hours and 10 minutes E. 4 hours and 33 minutes

A. 2 hours and 24 minutes EQUATION RELATIONSHIP Sally can paint 1/4 of the house in 1 hour. John can paint 1/6 of the same house in 1 hour. In order to determine how long it will take them to paint the house, when working together, the following equation may be written: 1/4 x+1/6 x=1. Solving for x gives 5/12 x=1, where x = 2.4 hours, or 2 hours, 24 minutes.

If Steven can mix 20 drinks in 5 minutes, Sue can mix 20 drinks in 10 minutes, and Jack can mix 20 drinks in 15 minutes, how much time will it take all 3 of them working together to mix the 20 drinks? A. 2 minutes and 44 seconds B. 2 minutes and 58 seconds C. 3 minutes and 10 seconds D. 3 minutes and 26 seconds E. 4 minutes and 15 seconds

A. 2 minutes and 44 seconds The amount of time it takes the three of them to mix the 20 drinks may be represented by the equation, 1/5+1/10+1/15=1/t, where t represents the number of minutes. Solving for t gives t=30/11, which equals 2.73 minutes. There are 60 seconds in a minute, so multiply 60 by 2.73 minutes to get 163.8 seconds. Divide that by 60, and it comes to approximately 2 minutes and 44 seconds.

If a record spins at the rate of 150° per second, how many complete revolutions does it make in one minute? A. 25 B. 144 C. 150 D. 9000

A. 25 This is a revolution problem and if it spins 150 degrees in one second we simply find the number of degrees that represents a minute. Then we will divide the total degrees by 360 degrees since there are 360 degrees in one revolution. 150 x 60 = 9000 9000/360 = 25

A package is dropped from an airplane. The height of the package at any time t is described by the equation, y(t)= -1/2 at2 + h0 where y is the height,h0 is the original height, and a is the acceleration due to gravity. The value of a is 32ft/sec2. If the airplane is flying at 30,000 feet, what is the altitude of the package 15 seconds after it is dropped? A. 26400 ft B. 22800 ft C. 15640 ft D. 7200 ft E. 0 ft

A. 26400 ft Simply evaluating the expression yields y(15)=-1/2*32*(15)2+30,000=-16*225+30,000=-3600+30,000=26400. Since this value is unique, all the other answers are incorrect.

What is the slope of the line whose equation is 3x-4y-16=0? A. 3/4 B. 4/3 C. 3 D. -4

A. 3/4 Re-writing the equation in "y=" form gives y=(3/4)x-4. When an equation is in "y=" form, the slope is the number in front of the x.

1. If Lynn can type a page in p minutes, what piece of the page can she do in 5 minutes? A. 5/p B. p - 5 C. p + 5 D. p/5 E. 1- p + 5

A. 5/p PROPORTIONS The following proportion may be written: 1/p=x/5. Solving for the variable, x, gives xp = 5, where x=5/p. So, Lynn can type 5/p pages, in 5 minutes.

THE PERCENT INCREASE The last week of a month a car dealership sold 12 cars. A new sales promotion came out the first week of the next month and the sold 19 cars that week. What was the percent increase in sales from the last week of the previous month compared to the first week of the next month? A. 58% B. 119% C. 158% D. 175% E. 200%

A. 58% The percent increase may be represented as (19-12)/12, which equals 0.583 ?. Thus, the percent of increase was approximately 58%.

. If Leah is 6 years older than Sue, and John is 5 years older than Leah, and the total of their ages is 41. Then how old is Sue? A. 8 B. 10 C. 14 D. 19 E. 21

A. 8 Three equations may initially be written to represent the given information. Since the sum of the three ages is 41, we may write, l + s + j = 41, where l represents Leah's age, s represents Sue's age, and j represents John's age. We also know that Leah is 6 years older than Sue, so we may write the equation, l = s + 6. Since John is 5 years older than Leah, we may also write the equation, j = l + 5. The expression for l, or s + 6, may be substituted into the equation, j = l + 5, giving j = s + 6 + 5, or j = s + 11. Now, the expressions for l and j may be substituted into the equation, representing the sum of their ages. Doing so gives: s + 6 + s + s + 11 = 41, or 3s = 24, where s = 8. Thus, Sue is 8 years old.

If r = 5 z then 15 z = 3 y, then r = A. y B. 2 y C. 5 y D. 10 y E. 15 y

A. y The value of z may be determined by dividing both sides of the equation, r=5z, by 5. Doing so gives r/5=z. Substituting r/5 for the variable, z, in the equation, 15z=3y, gives 15(r/5)=3y. Solving for y gives r = y.

You purchase a car making a down payment of $3,000 and 6 monthly payments of $225. How much have you paid so far for the car? A. $3225 B. $4350 C. $5375 D. $6550 E. $6398

B. $4350 The amount you have paid for the car may be written as $3,000 + 6($225), which equals $4,350.

Lee worked 22 hours this week and made $132. If she works 15 hours next week at the same pay rate, how much will she make? A. $57 B. $90 C. $104 D. $112 E. $122

B. $90 The following proportion may be used to determine how much Lee will make next week: 22/132=15/x. Solving for x gives x = 90. Thus, she will make $90 next week, if she works 15 hours.

If Sam can do a job in 4 days that Lisa can do in 6 days and Tom can do in 2 days, how long would the job take if Sam, Lisa, and Tom worked together to complete it? A. 0.8 days B. 1.09 days C. 1.23 days D. 1.65 days E. 1.97 days

B. 1.09 days The amount of time it will take the three of them to finish the job, when working together, may be modeled by the equation, 1/4+1/6+1/2=1/t, where t represents the number of days. Solving for t gives t=12/11, or 1.(09). Thus, it will take the three of them 1.09 days to finish the job

TRICKY ONE Jim can fill a pool carrying buckets of water in 30 minutes. Sue can do the same job in 45 minutes. Tony can do the same job in 1 ½ hours. How quickly can all three fill the pool together? A. 12 minutes B. 15 minutes C. 21 minutes D. 23 minutes E. 28 minutes

B. 15 minutes The amount of time it takes the three of them to fill the pool may be represented by the equation, 1/30+1/45+1/90=1/t, where t represents the number of minutes. Solving for t gives t = 15. Thus, after 15 minutes, the three of them will fill the pool, when working together.

SUPER IMPORTANT ONE Two cyclists start biking from a trail's start 3 hours apart. The second cyclist travels at 10 miles per hour and starts 3 hours after the first cyclist who is traveling at 6 miles per hour. How much time will pass before the second cyclist catches up with the first from the time the second cyclist started biking? A. 2 hours B. 4 ½ hours C. 5 ¾ hours D. 6 hours E. 7 ½ hours

B. 4 ½ hours The intersection of the graphs of the equations, y = 6x and y = 10x - 30, represents the time (x) and distance (y), where the second cyclist catches up with the first cyclist. The point of intersection is (7½, 45). Thus, after 7½ hours from the time the first cyclist starts and 4½ hours from the time the second cyclist starts, the second cyclist catches up with the first cyclist.

What simple interest rate will Susan need to secure to make $2,500 in interest on a $10,000 principal over 5 years? A. 4% B. 5% C. 6% D. 7% E. 8%

B. 5% Simple interest is represented by the formula, I = Prt, where I represents the interest amount, P represents the principal, r represents the interest rate, and t represents the time. Substituting 2,500 for I, 10,000 for P, and 5 for t, gives the equation, 2,500 = 10,000(r)(5). Thus, r = 0.05, or 5%.

. You need to purchase a textbook for nursing school. The book cost $80.00, and the sales tax where you are purchasing the book is 8.25%. You have $100. How much change will you receive back? A. $5.20 B. $7.35 C. $13.40 D. $19.95 E. $21.25

C. $13.40 The amount you will pay for the book may be represented by the expression, 80+(80*0.0825). Thus, you will pay $86.60 for the book. The change you will receive is equal to the difference of $100 and $86.60, or $13.40.

Simon arrived at work at 8:15 A.M. and left work at 10: 30 P.M. If Simon gets paid by the hour at a rate of $10 and time and ½ for any hours worked over 8 in a day. How much did Simon get paid? A. $120.25 B. $160.75 C. $173.75 D. $180 E. $182.50

C. $173.75 From 8:15 A.M. to 4:15 P.M., he gets paid $10 per hour, with the total amount paid represented by the equation, $10*8=$80. From 4:15 P.M. to 10:30 P.M., he gets paid $15 per hour, with the total amount paid represented by the equation, $15*6.25=$93.75. The sum of $80 and $93.75 is $173.75, so he was paid $173.75 for 14.25 hours of work.

Jim has 5 pieces of string. He needs to choose the piece that will be able to go around his 36-inch waist. His belt broke, and his pants are falling down. The piece needs to be at least 4 inches longer than his waist so he can tie a knot in it, but it cannot be more that 6 inches longer so that the ends will not show from under his shirt. Which of the following pieces of string will work the best? A. 3 feet B. 3 ¾ feet C. 3 ½ feet D. 3 ¼ feet E. 2 ½ feet

C. 3 ½ feet The inequality, 40 ≤ x ≤ 42, represents his situation. A length of 3 1/2 feet equals 42 inches, which satisfies the inequality

A student receives his grade report from a local community college, but the GPA is smudged. He took the following classes: a 2 hour credit art, a 3 hour credit history, a 4 hour credit science course, a 3 hour credit mathematics course, and a 1 hour science lab. He received a "B" in the art class, an "A" in the history class, a "C" in the science class, a "B" in the mathematics class, and an "A" in the science lab. What was his GPA if the letter grades are based on a 4 point scale? (A=4, B=3, C=2, D=1, F=0) A. 2.7 B. 2.8 C. 3.0 D. 3.1 E. 3.2

C. 3.0 The GPA may be calculated by writing the expression, ((3*2)+(4*3)+(2*4)+(3*3)+(4*1))/13, which equals 3, or 3.0.

If 8x + 5x + 2x + 4x = 114, the 5x + 3 = A. 12 B. 25 C. 33 D. 47 E. 86

C. 33 The given equation should be solved for x. Doing so gives x = 6. Substituting the x-value of 6 into the expression, 5x + 3, gives 5(6) + 3, or 33.

If two planes leave the same airport at 1:00 PM, how many miles apart will they be at 3:00 PM if one travels directly north at 150 mph and the other travels directly west at 200 mph? A. 50 miles B. 100 miles C. 500 miles D. 700 miles E. 1,000 miles

C. 500 miles The Pythagorean theorem may be used to solve the problem. The vertical distance of the plane traveling north, after 2 hours, is 300 miles. The horizontal distance of the plane traveling west, after 2 hours, is 400 miles. Thus, the following equation represents the distance between the planes, at 3 P.M.: 3002+4002=c2. Solving for c gives √250,000=c, or c = 500. After 2 hours, the planes are 500 miles apart.

If y = 3, then y3(y3-y)= A. 300 B. 459 C. 648 D. 999 E. 1099

C. 648 Substituting 3 for y gives 33 (33-3), which equals 27(27 - 3), or 27(24). Thus, the expression equals 648

A movie theater sells 3 sizes of popcorn (small, medium, and large) with 3 choices of toppings (no butter, butter, extra butter). How many possible ways can a bag of popcorn be purchased? A. 1 B. 3 C. 9 D. 27

C. 9 3 x 3 = 9

The sales price of a car is $12,590, which is 20% off the original price. What is the original price? A. $14,310.40 B. $14,990.90 C. $15,290.70 D. $15,737.50 E. $16,935.80

D. $15,737.50 $12,590 = Original Price - 0.2(Original Price) = 0.8(Original Price), Original Price = $12,590/0.8 = $15,737.50

3. Employees of a discount appliance store receive an additional 20% off of the lowest price on an item. If an employee purchases a dishwasher during a 15% off sale, how much will he pay if the dishwasher originally cost $450? A. $280.90 B. $287 C. $292.50 D. $306 E. $333.89

D. $306 Sale Price = $450 - 0.15($450) = $382.50, Employee Price = $382.50 - 0.2($382.50) = $306

Grace has 16 jellybeans in her pocket. She has 8 red ones, 4 green ones, and 4 blue ones. What is the minimum number of jellybeans she must take out of her pocket to ensure that she has one of each color? A. 4 B. 8 C. 12 D. 13 E. 16

D. 13 If she removes 13 jellybeans from her pocket, she will have 3 jellybeans left, with each color represented. If she removes only 12 jellybeans, green or blue may not be represented.

The city council has decided to add a 0.3% tax on motel and hotel rooms. If a traveler spends the night in a motel room that costs $55 before taxes, how much will the city receive in taxes from him? A. 10 cents B. 11 cents C. 15 cents D. 17 cents E. 21 cents

D. 17 cents The amount of taxes is equal to $55*0.003, or $0.165. Rounding to the nearest cent gives 17 cents.

A number x can be expressed as 210 more than 30 percent of itself. What is the value of x? A. 30 B. 63 C. 210 D. 300

D. 300 Remember when you see word equations to get the numbers out of the words. We substitute and distribute. x = .3x + 210 .7x = 210 x = 300

If the average of three numbers is V. If one of the numbers is Z and another is Y, what is the remaining number? A. ZY - V B. Z/V - 3 - Y C. Z/3 - V - Y D. 3V- Z - Y E. V- Z - Y

D. 3V- Z - Y The average of the three numbers may be written as (Z+Y+x)/3=V, where x represents the value of the third number. Solving for x will give the value of the remaining number. Multiplying both sides of the equation by 3 gives Z + Y + x = 3V. Subtraction of Z and Y, from both sides of the equation gives x = 3V - Z - Y. The value of the remaining number is 3V - Z - Y.

A study reported that in a random sampling of 100 women over the age of 35 showed that 8 of the women were married 2 or more times. Based on the study results, how many women in a group of 5,000 women over the age of 35 would likely be married 2 or more times? A. 55 B. 150 C. 200 D. 400 E. 600

D. 400 The following proportion may be used to solve the problem: 8/100=x/5000. Solving for x gives x = 400. Thus, 400 women, out of the random sample of 5,000, will likely have been married 2 or more times.

Alfred wants to invest $4,000 at 6% simple interest rate for 5 years. How much interest will he receive? A. $240 B. $480 C. $720 D. $960 E. $1,200

E. $1,200 Simple interest is represented by the formula, I = Prt, where P represents the principal amount, r represents the interest rate, and t represents the time. Substituting $4,000 for P, 0.06 for r, and 5 for t gives I = (4000)(0.06)(5), or I = 1,200. So, he will receive $1,200 in interest.

A tire on a car rotates at 500 RPM (revolutions per minute) when the car is traveling at 50 km/hr (kilometers per hour). What is the circumference of the tire, in meters? A. 50,000/2π B. 50,000/(60*2π) C. 50,000/(500*2π) D. 50,000/60 E. 10/6

E. 10/6 It is not necessary to use the circle circumference formula to solve the problem. Rather, note that 50 km/hr corresponds to 50,000 meters per hour. We are given the car tire's revolutions per minute and the answer must be represented as meters; therefore, the speed must be converted to meters per minute. This corresponds to a speed of 50,000/60 meters per minute, as there are 60 minutes in an hour. In any given minute, the car travels 50,000/60 meters/min, and each tire rotates 500 times around, or 500 times its circumference. This corresponds to 50,000/(60�-500)=10/6 meters per revolution, which is the circumference of the tire.

John is traveling to a meeting that is 28 miles away. He needs to be there in 30 minutes. How fast does he need to go to make it to the meeting on time? A. 25 mph B. 37 mph C. 41 mph D. 49 mph E. 56 mph

E. 56 mph The following equation may be used to find the speed at which he needs to travel: 28/x=1/2. Thus, x = 56. He needs to travel 56 mph, in order to make it to the meeting on time.

(2a^2b - 3c^3)(3a^3b + 4c) = A. 5a^6b^2 + 12c^4 - 9a^3bc^3 - 12c^4 B. 5a^5b^2 + 8a^2bc - 9a^3bc^3 + 12c^4 C. 6a^5b2 + 8a^2bc - 9a3bc3 + 12c4 D. 6a^6b2 + 8a^2bc - 9a^3bc^3 - 12c4 E. 6a^5b2 + 8a^2bc - 9a^3bc^3 - 12c4

E. 6a^5b2 + 8a^2bc - 9a^3bc^3 - 12c4 To multiply two binomials, use the FOIL method. FOIL stands for First, Outside, Inside, Last. When multiplying each pair of terms, remember to multiply the coefficients, then add the exponents of each separate variable. So, the product of the First terms is 2a2 b*3a3b = 6a5b2. The product of the Outside terms is 2a2 b*4c=8a2 bc. The product of the Inside terms is -3c3*3a3 b=-9a3 bc3 . The product of the Last terms is-3c3*4c=-12c4. The final answer is simply the sum of these four products.

During a 4-day festival, the number of visitors tripled each day. If the festival opened on a Thursday with 345 visitors, what was the attendance on that Sunday? A. 345 B. 1,035 C. 1,725 D. 3,105 E. 9,315

E. 9,315 The problem represents a geometric sequence, with a common ratio of 3. Thus, the problem may be modeled with the equation, a4=345.34-1, where a4=9,315. The problem may also be solved by writing the sequence, 345, 1035, 3105, 9315, and identifying the value of the fourth term as the number in attendance for Sunday, or the fourth day.

Which of the following is not a rational number? A. -4 B. 1/5 C. 0.8333333... D. 0.45 E. square root 2

E. square root 2 has a decimal expansion that does not terminate or repeat (1.414213562...). Thus, it is an irrational number

Mary is reviewing her algebra quiz. She has determined that one of her solutions is incorrect. Which one is it? A. 2x + 5 (x-1) = 9, x = 2 B. p - 3(p-5) = 10, p = 2.5 C. 4 y + 3 y = 28, y = 4 D. 5 w + 6 w - 3w = 64, w = 8 E. t - 2t - 3t = 32, t = 8

E. t - 2t - 3t = 32, t = 8 The correct solution is t = -8. When adding t to -5t, it looks like she forgot to include the negative sign on 4t, which gave an incorrect solution of positive 8.

Which of the following expressions is equivalent to (a+b)(a-b) ? a^2-b^2 (a+b)^2 (a-b)^2 ab(a-b) ab(a+b)

a^2-b^2 Compute the product using the FOIL method, in which the First terms, then the Outer terms, the Inner terms, and finally the Last terms are figured in sequence of multiplication. As a result (a+b)(a-b)=a2-ab+ba-b2,. Since ab is equal to ba, the middle terms cancel out each other which leaves a2- b2.