PCAT Quantitative Reasoning

Which of the following expressions is equal to log(81) + log(729)? A. 6 log(3) B. 10 log(3) C. 6 log(10) D. 10 log(10)

b. 10 log(3)

A number is picked randomly between 1 and 10. If this is done twice, what is the probability that both the sum and product of the numbers chosen is odd? a. 0 b. 1/2 c. 2/3 d. 1

(A) In order for the sum of two numbers to be odd, one of the number must be odd and the other must be even. In order for the product of two numbers to be odd, they must both be odd. Thus, it is impossible for any two numbers to have an odd sum and product. So the probability is 0.

A student's average (arithmetic mean) score on 5 tests is 72. If the lowest score is dropped, the average will increase to 84. What is the lowest score? a. 18 b. 24 c. 32 d. 34

(B) The student's average score is the sum of all the scores divided by the number of tests. If the average is 72 for 5 tests, then the sum of all the scores is 5 × 72 or 360. The average of the 4 highest scores is 84; the sum of these scores is 4 × 84 or 336. The difference between the sum of all 5 scores and the sum of the 4 highest scores is just the lowest score. Therefore, the lowest score is 360 - 336 or 24 (admittedly, not such a good mark!).

Which of the following functions has a finite limit as →∞? A. sin x B. tan x C. e^-x D. 1 - e^x

(C) e^-x Trigonometric functions are periodic. They repeat their pattern forever, so they do not have a limit as x approaches infinity. Choice (C) can be rewritten as 1/e^x As x increases, the denominator approaches infinity, so the function approaches 0. This is a finite limit. Choice (D) approaches negative infinity as x approaches infinity.

A crate of apples contains 1 bruised apple for every 30 apples in the create. Three out of every four bruised apples are considered not fit to sell, and every apple that is not fit to sell is bruised. If there are 12 apples not fit to sell in the crate, then how many total apples are there in the crate?

480. Strategy: backsolving with answer choices.

Company Z spent 1/4 of its revenues last year on marketing and 1/7 of the remainder on maintenance of its facilities. What fraction of last year's original revenues did company z have left after its marketing and maintenance expenditures?

9/14 Company Z spends portions of its revenues on two things. It is important to note that the answer choices are fractions and represent the fraction of last year's revenues that company Z had left after the expenses. Notice that you are given absolutely no way to know how much revenue company Z received last year, so you can use Picking Numbers to determine the total revenue. Since you must take 1/4 and 1/7 of the revenue, a number divisible by both 4 and 7 will be most manageable. Make the calculations as easy as possible by picking the lowest common multiple of 4 and 7: 28. Now "Company Z spent 1/4 of its revenues last year on marketing," becomes "Company Z spent $7 last year on marketing." The next sentence mentions "the remainder," so you need to know what that is. Since $28 - $7 = $21, "Company Z spent 1/7 of the remainder on maintenance," means "Company Z spent $3 on maintenance." What's left from $28 after company Z spent $7 and then $3 is $18. So the answer is 18/28 = 9/14

A 3-liter insecticide solution contains 90 percent inert ingredients by volume. How many additional liters of inert ingredients must be added to yield a solution which contains 95 percent inert ingredients by volume? A. 0.15 B. 1.5 C. 3 D. 4.5

C. 3 In this question, we are told that 3 liters of an insecticide solution is composed of 90 percent inert ingredients. We are asked to find the number of liters of inert solution that need to be added to the 90 percent solution to get a solution of 95 percent inert ingredients. If the initial solution is 90 percent inert ingredients, then it has 10% active ingredients. When adding more inert ingredients to the solution, the percent of active ingredients will decrease. It is important to note that the total amount of insecticide will stay the same; however, its concentration will decrease. In order for the strength of the insecticide solution to decrease to half its original strength, there must be twice as much solution. If we started with 3 liters of the 10% insecticide solution, then by adding 3 liters of inert ingredients, our solution will contain 95% inert ingredients.

A solution is 90 percent glycerin. If there are 4 gallons of the solution, how much water, in gallons, must be added to make a 75-percent glycerin solution? A. 1.4 B. 1.2 C. 1.0 D. 0.8

We have 4 gallons of 90% glycerin; we want to know how much water we must add to decrease the glycerin concentration to 75%. 4 gallons of a 90% glycerin mixture has 9/10 × 4 or 3.6 gallons of glycerin. At what volume of solution will this 3.6 gallons of glycerin represent 75% of the solution? Set up an equation: 3.6 gallons = 75% of x gallons 3.6 = 3/4x x = 4/3 * 3.6 = 4.8 So we will have a 75% solution when there are 4.8 gallons of liquid. Since we started with 4 gallons, we need to add 4.8 - 4 or 0.8 gallons of water.

A code is to be made by arranging 7 letters. Three of the letters used will be the letter A, two of the letters used will be the letter B, one of the letters used will be the letter C, and one of the letters used will be the letter D. If there is only one way to present each letter, how many different codes are possible? a. 42 b. 210 c. 420 d. 5,040

You need to make a seven-letter code, but some of the letters are repeated. You have three As, two Bs, one C, and one D. Calculate the number of different permutations, remembering to take the repeated letters into account. To calculate the number of permutations where some of the elements are indistinguishable, divide the total number of permutations (hence, 7!) by the factorial of the number of indistinguishable elements. So you have 7!/3!2! = (7 * 6 * 5 * 4 * 3 * 2 * 1) / (3 * 2 * 1)(2 * 1) = 420.

If bear claw doughnuts cost x cents per dozen, how much will 24 bear claws cost in dollars? a. $x/50 b. $x/25 c. $25x/100 d. $25x/50

a. $x/50 The answer choices are expressions with variables, so anticipate using algebra. This is a unit conversion problem requiring you to convert bear claws to dollars. 24 bear claws is the same as 2 dozen bear claws. If 12 (one dozen) bear claws cost x cents, then 24 bear claws will cost 2x cents. To convert 2xcents to dollars, divide by 100. 2x/100 = x/50 (A) best matches the prediction.

For the function y = 2x2 + 2, at which point is the slope 8? a. (2, 10) b. (10, 2) c. (4, 34) d. (34, 4)

a. (2, 10) First find the derivative of y = f(x) = 2x2 + 2. The slope is the derivative evaluated at a certain x value. f'(x) = 2(2x) + 0 = 4x. To find the point that has a slope of 8, set f'(x) equal to 8 and solve for x: 4x = 8 x = 2. To find the y value for x = 2, plug x = 2 into the equation for y. For x = 2, y = 2(22) + 2 = 8 + 2 = 10. The point with slope 8 is (2, 10). (A) is correct.

If y is a positive integer and 3y + 2x < 5, which of the following is a possible value for x? a. 0 b. 1 c. 2 d. 3

a. 0 Solve the inequality for x. 3y + 2x < 5 2x < 5 - 3y x < 5/2 - 3/2y If y is a positive integer, then the smallest value y can have is 1. When y = 1: x < 5/2 - 3/2(1) x < 1 So any correct value of x must be less than 1. The only value less than 1 is 0, (A).

If log(144) = 2.16, then what is log(12)? A. 1.08 B. 1.16 C. 2.02 D. 3.46

a. 1.08

A fair six-sided die with sides numbered 1, 2, 3, 4, 5, and 6 is thrown twice. What is the probability that both the sum and the product of the two numbers thrown will be even? a. 1/4 b. 1/2 c. 4/9 d. 1/3

a. 1/4 Let's keep in mind that the numbers on the dice are integers. In order for the sum of the integers to be even, both integers must be even or both integers must be odd. In order for the product of the two integers to be even, at least one of the integers must be even. To meet both of these conditions, both numbers must be even. The probability formula is probability = (# of favorable outcome)/(# of possible outcomes) On each die there are the 3 even numbers 2, 4, and 6 among the first 6 positive integers. The probability that the first number is even is 3/6, reduced to 1/2. The results of throwing each die are independent of one another. The probability that the second number is even is again 3/6, reduced to 1/2. The probability that both the sum and the product of the two numbers is even, which is the probability that both numbers are even, is 1/2 * 1/2 = 1/4

The ratio of Democrats to Republicans in a district of 6 : 5. The ratio of Republicans to Independents is 3 : 1. What is the ratio of Democrats to Independents in the district? a. 18 : 5 b. 9 : 5 c. 6 : 5 d. 2 : 5

a. 18 : 5 In this question, we are asked to calculate the ratio of Democrats to Independents given that the ratio of Democrats to Republicans is 6:5 and the ratio of Republicans to Independents is 3:1. There are essentially two fractions in this question: Democrats to Republicans = 6/5 and Republicans to Independents = 3/1. The common piece of these two ratios is Republicans. In order to relate Democrats to Independents, we have to make the Republican term the same in each ratio. In other words, we need to find a multiple of 5 and 3, which is 15. The Democrat to Republican ratio now reads 18:15 and the ratio of Republicans to Independents is 15:5. For every 15 Republicans there are 18 Democrats and 5 Independents. The ratio of Democrats to Independents is 18:5.

For all x, 2^x + 2^x + 2^x + 2^x = a. 2^(x+2) b. 2(x+4) c. 2^3x d. 2^4x

a. 2^(x+2) The sum of 4 identical quantities is 4 times one of those quantities, so the sum of the four terms 2^x is 4 times 2^x: 2^x + 2^x + 2^x + 2^x = 4(2^x) = 2^2(2^x) = 2^(x+2)

When 3x^3 - 7x + 7 is divided by x+2, the remainder is: a. -5 b. -3 c. 1 d. 3

b. -3 Use long division. Watch out: The expression that goes under the division sign needs a place-holding 0x^2 term

A tired mathematician only takes her coffee in a specific 7:4:3 ratio of black coffee to milk to honey. On Monday she drank 5 cups of coffee mixed this way. How many cups of milk did she use? a. 4/7 b. 10/7 c. 4/11 d. 2/7

b. 10/7 (B) Choice B is the correct answer. The question states that the mathematician consumed a total of 5 cups of coffee. We need to determine what fraction of those 5 cups is composed of milk. The total number of parts of liquid per cup of coffee is 14, because the ratio of black coffee to milk to honey is 7 : 4 : 3, which can be considered 7 + 4 + 3 = 14. Since there are 4 parts of milk per cup of coffee, the total fraction of milk per cup is 4/14 , therefore 4/14 of 5 cups is 20/14 which simplifies to 10/7.

In a certain state lottery, 40 percent of the money collected goes towards education. If during a certain week 6.4 million dollars were obtained for education, how much was collected during that week, in millions of dollars? a. 25.6 b. 16.0 c. 8.96 d. 8.0

b. 16.0 You're told that 40% of the total money collected for the lottery is 6.4 million dollars, and you're asked to find the total. Write this information as an equation in terms of millions: 40% 3 total = 6.4 4/10 * total = 6.4 total = 10/4 * 6.4 total = 64/4 = 16 You could have worked from the answer choices, multiplying each by 40% to see which gives you 6.4. You also might have realized that 40% is a little less than one-half and picked 16 as the answer choice that's a bit more than 12.8, or twice 6.4.

Brand A coffee costs twice as much as brand B coffee. If a certain blend is brand A and brand B, what fraction of the cost of the blend is brand A ? a. 1/3 b. 2/5 c. 1/2 d. 2/3

b. 2/5 The question is asking for the cost of brand A within the blend relative to the cost of the whole blend. The simplest approach here is to pick numbers for the prices of the two brands of coffee. Suppose that the cost of brand B is $4 per pound. Picking 4 will make for easy calculations since the amounts used for the blend are expressed in fourths. Since brand A costs twice as much as brand B, brand A costs 2 × $4 = $8 per pound. Each pound of the blend contains 1/4 pound of brand A for a cost of 1/4 * $8 = $2, and 3/4 pound of brand B for a cost of 3/4 * $4 = $3. So the total cost for a pound of this blend is $2 + $3 = $5. Brand A's cost represents $2/$5, or 2/5 of the total cost.

If (3^x^2)(9^x)(3) = 27 and x>0, what is the value of x? a. 0.27 b. 0.41 c. 0.73 d. 1.41

c. 0.73 Watch what happens when you express everything as powers with a base of 3: (3^x^2)(3^2)^x (3^1) = 3^3 The left side of the equation is the product of powers with the same base, so just add the exponents: 3^(x^2 + 2x + 1) = 3^3 Now the two sides of the equation are powers with the same base, so you can just set the exponents equal: x^2 + 2x + 1 = 3 x = -1 +/- sqrt(3) The positive value is -1 + sqrt(3), which is approximately 0.732.

At x = 3, what is the slope of the function y = 4x3 + 2x + 4? a. 18 b. 54 c. 110 d. 118

c. 110 To find the slope of the function, you need to find the derivative of the function. Remember the rule for derivatives: If y = xn, then the derivative is nxn - 1. The derivative of 4x3 + 2x + 4 is: 4(3x2) + 2(1) + 0 = 12x^2 + 2. Evaluate the derivative by plugging in x = 3: 12(32) + 2 = 12(9) + 2 = 108 + 2 = 110.

If (2 + xi)(4 + 3i) = 2 + 14i, then what is the value of x? a. -2 b. 0 c. 2 d. 3

c. 2 To solve this equation, first distribute the parenthesis and group like terms: (2 + xi)(4 + 3i) = 2 + 14i 8 + 6i + 4xi + 3xi2 = 2 + 14i Now use the fact that i2 = -1: 8 + 6i + 4xi + 3x(-1) = 2 + 14i 8 + 6i + 4xi- 3x = 2 + 14i This may look complicated, but remember that when adding complex numbers, you can separate the real part and the imaginary part. (8 - 3x) + (4x + 6)i = 2 + 14i This tells us that 8 - 3x = 2 and 4x + 6 = 14. Solve either of these equations to get x: 8 = 3x + 2 -3x = -6 x = 2

A certain school has 50 students assigned to five distinct classes so that the numbers of students in the classes are consecutive and each student is assigned to only one class. What is the probability that a student selected at random from the 50 students is in one of the two largest classes? a. 38% b. 42% c. 46% d. 48%

c. 46% To find the probability, you need possible outcomes and desired outcomes. The possible outcomes here are just the total number of students you have to choose from. The desired outcomes are the number of students in the two largest classes. How can you find that, though? The key to this problem is the fact that the numbers of students in the classes are consecutive. Whenever you see consecutive in a math problem, it's there for a specific reason. It tells you that each number in the series separated from the next by a fixed amount. Consecutive integers are separated by 1, consecutive even integers by 2, etc. So here you know that if you add up the numbers of students in all the classes, you'll get 50. You also know that the numbers of students in the classes are separated by 1. So let's call the number of students in the smallest class x. The next largest class would have x + 1 students, then x +2 students, x + 3 students, and x + 4 students. If you add these up, you get 5x + 10 = 50. Now you can solve for x: subtract 10 from both sides to get 5x = 40, then divide both sides by 5 to get x = 8. So the smallest class has 8 students, and the two largest classes then will have 8 + 3 and 8 + 4 students respectively, which gives a total of 23 students in the two largest classes. So the probability is 23/50 = 23/50 * 2/2 = 46/100 = 46%

What is the range of the function f(x) = 1 -e-2x if the domain may only contain real numbers? A. (-∞, ∞) B. (-∞, 0) C. (1, ∞) D. (-∞, 1)

d. ( -infinity, 1)

A six-sided die is thrown twice. What is the probability that the product of the two numbers thrown will be odd? a. 1/2 b. 4/9 c. 1/3 d. 1/4

d. 1/4 For both the product of two integers to be odd, both integers must be odd. There is a 3 in 6 chance for an odd number to occur on any die thrown. Since both dice must have an odd number, you combine the probabilities by multiplying them. 1/2 * 1/2 = 1/4

List L consists of terms {0, 4, 9, 11, 17, 23} and has a standard deviation of a. When one of the terms is removed from List L, the resulting set of terms is List M, which has a standard deviation of b. The value of b - a would be greatest if which of the following numbers were removed from List L to form List M? a. 0 b. 4 c. 9 d. 11 e. 23

d. 11 This question tests our understanding of standard deviation. Standard deviation is a measure of dispersal. Roughly speaking, the more "tightly-packed" a set of numbers is, the smaller the standard deviation will be. Conversely, the more "spread out" a set of numbers is, the larger the standard deviation will be. Specifically, the question asks for the largest possible value of the quantity b - a. Since b and a represent the standard deviations of Lists M and L, respectively, we can maximize b - a by making the standard deviation of M as large as possible relative to the standard deviation of L. Lists M and L are very similar: the only difference is that one term is removed from List L to form List M. So the question is really asking, "What number can we remove that will increase the standard deviation as much as possible?" There is no need to calculate the standard deviation of either list. Instead, we need to think critically. If we take away one of the "extreme" values (0 or 23) from List L, the remaining numbers will be more compact than those in the original list, and the standard deviation will decrease. On the other hand, if we remove one of the "middle" values from List L, the remaining numbers will be more spread out than those in the original list, and the standard deviation will increase. Since our goal is to maximize b - a, we should make b, the standard deviation of List M, as large as possible. Therefore, the correct answer will be the number in List L that is the closest to the mean. Therefore, we should calculate the mean of List L by calculating the sum of its elements and dividing by 6, the number of terms. The sum is 0 + 4 + 9 + 11 + 17 + 23 = 64, so the mean is 64 divided by 6, or approximately 10.67. Since 11 is the number that is closest to 10.67, Answer Choice (D) is correct

The ratio of new cars to old cars in a certain city subway system is 5:3. If 9/10 of the new cars are considered "graffiti-free," which of the following is the largest possible ratio of graffiti-free cars to graffiti-covered cars in the entire system? a. 15 : 16 b. 45 : 35 c. 9 : 1 d. 15 : 1

d. 15 : 1 To get the largest possible ratio of graffiti-free cars to graffiti-covered cars, maximize the number of graffiti-free cars and minimize the number of graffiti-covered cars. Do this by considering as graffiti-covered only the cars you are explicitly told are not graffiti-free: the 1/10 of the new cars. Assume that all the other cars are graffiti-free, including all of the old cars, since you aren't told that they have to be graffiti-covered. Before calculating, notice that this will make the largest possible ratio much greater than 1, that is, the numerator (or the first number in the ratio) will be much larger than the denominator (or the second number in the ratio). So you can eliminate (A) and (B). The ratio of new to old cars is 5:3, so 5/5+3 or 5/8 of the total cars are new. Only 1/10 of the new cars have graffiti, so a minimum of 1/10 * 5/8 =1/16 of all the cars have graffiti. The rest of the cars, or 15/16 could be graffiti-free. So the largest possible ratio of graffiti-free to graffiti-covered cars is 15/16 : 1/16, or 15 : 1

Amanda goes to the toy store to buy 1 ball and 3 different board games. If the toy store is stocked with 3 types of balls and 6 types of board games, how many different selections of the 4 items can Amanda make? a. 20 b. 23 c. 40 d. 60

d. 60 Amanda is buying 1 ball and 3 different board games from a selection of 3 balls and 6 types of board games. You need to find the number of different selections of 4 items that Amanda can make. Amanda is choosing 3 board games from a total of 6, so you can use the combination formula (since a different arrangement of the same board games is not considered a different selection) to find the number of combinations of board games: nCk = n! / k!(n-k)! 6C3 = 6! / 3!(6-3)! 6C3 = 6 * 5 * 4 * 3! / 3! * 3! 6C3 = 20 Amanda has 20 ways to choose 3 board games from a total of 6. For each of those 20 ways, Amanda can choose 1 of 3 balls, so there are 20 * 3 = 60 different ways for Amanda to choose 1 ball and 3 games from 3 balls and 6 games.

If a is 40 percent of c, and b is 60 percent of c, what percent is a of b? A. 20% B. 24% C. 33 1/3% D. 66 2/3%

d. 66 2/3% Since both a and b are given in terms of c, pick a number for c. Since this is a percent problem, let c = 100. a is 40% of c and 40% of 100 = 40. b is 60% of c and 60% of 100 = 60. To find what percent a is of b, plug into the formula part/whole * 100%: a/b * 100% or 40/60 * 100% = 2/3 *100% = 66 2/3%

A machine is made up of two components, A and B. Each component either works or fails. The failure or non failure of one component is independent of the failure or nonfailure of the other component. The machine works if at least one of the components works. If the probability that each component works is 2/3, what is the probability that the machine works? a. 1/9 b. 4/9 c. 1/2 d. 8/9

d. 8/9 The fastest way to do this is to find the probability that neither component works and subtract that from 1. Since the probability of a component working is 2/3, the probability of a component not working is 1 - 2/3 = 1/3. Therefore, the probability that neither component works is 1/3 * 1/3 = 1/9, and the probability that the machine works is 1 - 1/9 = 8/9.

How many integers are in the solution set of |4x +3| < 8? a. none b. two c. three d. four

d. four If the absolute value of something is less than 8, then that something is between -8 and 8: |4x + 3| < 8 -8 < 4x + 3 < 8 -11 < 4x < 8 -11/4 < x < 5/4 There are four integers in that range: -2, -1, 0, and 1