QR Practice Test #10

A fair coin is flipped 5 times, what is the probability of getting a tail in the fifth flip? A. 1/2 B. 1/4 C. 1/6 D. 1/8 E. 1/16

A (This probability question involves coins. We are asked about the outcome of the fifth flip. Recall a fair coin has two sides: one side with heads and one side with tails. Therefore, there is a ½ or 50% chance of getting heads and a ½ or 50% chance of getting tails. We do not care about how many times the coin is flipped prior to the fifth flip because each flip is independent. Therefore, there is a ½ or 50% chance the fifth coin lands on tails, so choice A is the correct answer.)

If a coin is flipped 3 times, what is the probability of obtaining at least one tail? A. 7/8 B. 1/8 C. 15/16 D. 1/2 E. 1/16

A

What is the mean, median, mode, and range of the following set, in that order? {1, 2, 4, 5, 6, 6, 7, 9} A. 5, 5.5, 6, 8 B. 5.5, 5, 6, 9 C. 5.5, 5.5, 6, 9 D. 5, 5.5, 2, 8 E. 5, 6, 2, 8

A (STEP 1: Finding the range of a set is the fastest because we can quickly determine that the smallest and largest value of the set is [1, 9], so the range is 9 - 1 = 8. We can check that answer choices A, D, and E satisfy this and eliminate answer choices B and C. STEP 2: Finding the mode of a set is similarly quick because we can quickly determine that the most repeated value of the set is 6. We can check that, of answer choices A, D, and E, only answer choice A matches our value. Therefore, we can confirm that answer choice A is correct. The moral of the question is that, when taking timed exams, finding the most efficient way to solve a problem can save lots of time and energy. Solving for the mean (which is 5) and median (which is 5.5) requires some calculation. However, finding the mode and range is more direct, and was enough information to help us narrow our answer choices to the correct one.)

If Garry drives 50 kilometers in 30 minutes, approximately, what is his speed in miles per hour? A. 62 mph B. 74 mph C. 80 mph D. 100 mph E. 120 mph

A (To solve this, we will first convert the information into the desired units then use the distance formula to solve. STEP 1: First, we convert 50 km to miles. STEP 2: Next, we convert 30 minutes to hours. STEP 3: Finally, we use the distance formula to solve for speed)

What is the solution set of: |x + 3| ≤ 10 A. -13 ≤ x ≤ 7 B. 7 ≤ x ≤ 13 C. -13 < x < 7 D. x ≤ -13 or x ≥ 7 E. 3 ≤ x ≤ 10

A. -13 ≤ x ≤ 7 This problem involves absolute values. When dealing with absolute values, a good first step is to get rid of it. To do this, we recall the following absolute value rules: Rule #1: |x| = a ↔ x = ± a Rule #2: |x| ≤ a ↔ -a ≤ x ≤ a Rule #3: |x| ≥ a ↔ x ≤ -a or x ≥ a To solve this, we will first re-write the inequality without the absolute value then we'll isolate x to get our answer. STEP 1: Using rule #2, we write the inequality without the absolute value. |x + 3| ≤ 10-10 ≤ x + 3 ≤ 10Note:we combined the two inequalities as one inequality to make calculations easier.STEP 2:We subtract3from both inequalities to isolatex. When we do this, we get: -13 ≤ x ≤ 7Therefore, the correct answer is choiceA.

If Garry drives 50 kilometers in 30 minutes, approximately, what is his speed in miles per hour? A. 62 mph B. 74 mph C. 80 mph D. 100 mph E. 120 mph

A. 62 mph This question tests our ability to use the distance formula and convert between units. The distance formula is: Since the information given is in different units than what the answer should be in, we must use dimensional analysis to convert. The general setup for dimensional analysis is: To convert the units correctly, we must know the following conversion factors: To solve this, we will first convert the information into the desired units then use the distance formula to solve. STEP 1: First, we convert 50 km to miles. STEP 2: Next, we convert 30 minutes to hours. STEP 3: Finally, we use the distance formula to solve for speed. Therefore, the correct answer is choice A. SHORTCUT: To save time with writing out equations, it would be quick to write out all the conversions in one calculation line

Dana is 20 years younger than her mother. In 4 years, the mother will be twice as old as Dana. How old is the mother now? A. 30 B. 36 C. 40 D. 44 E. 60

B (This is a great example of an age question. It is very likely you will encounter at least one age question on the OAT so be sure to fully understand this concept. In order to solve these questions, we need to translate the given information into algebraic expressions. Then, we solve using substitution. STEP 1: To begin, we write our expressions. Let "D" be Dana's current age and "M" be her mother's current age. Since we know "Dana is 20 years younger than her mother", we write: (1) D = M - 20Since we know "In 4 years, the mother will be twice as old as Dana", we write: (2) M + 4 = 2D + 8STEP 2:Next, we substituteDfrom the first equation into the second equation. M + 4 = 2D + 8M + 4 = 2(M - 20) + 8STEP 3:Finally, we solve forM. M + 4 = 2(M - 20) + 8 M + 4 = 2M - 40 + 8 M + 4 = 2M - 32 4 = M - 32 M = 36)

is x < 0 ? 1) xy < 0 2) x|y| < 0 A. Statement 1) alone is sufficient but statement 2) alone is NOT sufficient to answer the question. B. Statement 2) alone is sufficient but statement 1) alone is NOT sufficient to answer the question. C. Both statements 1) and 2) together are sufficient to answer the question but neither statement is sufficient alone. D. Each statement alone is sufficient to answer the question. E. Statements 1) and 2) together are not sufficient to answer the question

B (STEP 1: First, we attempt to solve the problem using statement 1 alone. xy < 0From the statement, we know the product of the two numbers is negative. This can only be the case if one of the two numbers is negative and the other is positive. In this case,xcan be the negative number or the positive number since we don't have any information abouty. Therefore, STatement 1alone is not sufficient to solve the problem. Eliminate choices A and D. STEP 2: Next, we attempt to solve the problem using statement 2 alone. x |y| < 0The absolute value ofymeans thatywill always be positive, soxmust be negative for the product to be negative. This information is sufficient alone to solve the question)

If two fair dice are rolled, what is the probability of getting a sum of 8? A. 3/36 B. 5/36 C. 1/4 D. 5/12 E. 21/36

B (Probability questions involving dice are common on the OAT and this is a good example of such a question. This problem requires multiple steps to solve. Our approach is to first find how many ways we can roll two dice with a sum of 8 then divide that value by the total possible outcomes the two dice can give. Recall a fair dice has six sides with each having a number 1-6. STEP 1: First, we find how many ways we can roll two dice to give us a sum of 8. We show these conditions in the table below: From the table above, we know there are five ways we can roll two dice to get a sum of 8. STEP 2: Next, we determine the total number of possible outcomes two dice rolls can give. STEP 3: Finally, we determine the probability of rolling a sum of 8.)

Approximately, how many yards are there in 5,000 centimeters? A. 50 B. 55 C. 110 D. 500 E. 5500

B (This problem tests our ability to convert units. We can use dimensional analysis to ensure that we do not confuse the units during conversion. STEP 1: First, we convert centimeters to meters: STEP 2: Next, we convert meters to yards, keeping in mind that there are approximately 1.1 yards in a meter: Conversion between meters and yards is tested on the exam, so make sure you know them for test day.)

The scores on a math exam worth 100 points are normally distributed with a mean of 70 and a standard deviation of 8. If Allen's z-score on the exam is -1.5, what grade did he receive on the exam? A. 48 B. 58 C. 62 D. 72.5 E. 78

B (z = standard score (In this case, it is -1.5) x = observed value (In this case, it is Allen's exam score) μ = mean of the sample (In this case, it is 70) σ = standard deviation of the sample (In this case, it is 8))

If 2x + 3y = 7x - 4y, what is x/y? A. 5/7 B. 7/5 C. 3/5 D. -5/3 E. -7/5

B. (This algebra problem involves a 2-variable equation. Since we are only given one 2-variable equation and not two, we cannot solve for the individual values of x and y through substitution or elimination. However, we recall a rule that comes in handy here: If ab = cd, thena/c=d/bTo solve this question, we will re-write the given equation in the form of the rule above and easily solve forx/y. STEP 1: First, we subtract 2x from both sides. 2x + 3y = 7x - 4y3y = 5x - 4ySTEP 2:Then, we add4yto both sides. This will isolate the two variables. 7y = 5xSTEP 3:Finally, our equation is now written in the form of the rule mentioned earlier, so we set up a ratio: 7y = 5xx/y=7/5)

If 5x + 3y = 15 and 3x + y = 4, what is the value of 2x + 2y? A. 10 B. 11 C. 12 D. 13 E. 16

B. (This algebra question involves a set of 2- variable equations. We typically solve these types of problems with substitution or elimination, but we notice if we subtract the equations from each other, we will get quickly get our answer. STEP 1: First, we line up both equations. STEP 2: Then, we subtract equation 2 from equation 1. To do this, we first distribute a minus sign for the second equation and add both equations together. Therefore, choice B is correct.)

Dana is 20 years younger than her mother. In 4 years, the mother will be twice as old as Dana. How old is the mother now? A. 30 B. 36 C. 40 D. 44 E. 60

B. 36 This is a great example of an age question. It is very likely you will encounter at least one age question on the OAT so be sure to fully understand this concept. In order to solve these questions, we need to translate the given information into algebraic expressions. Then, we solve using substitution. STEP 1: To begin, we write our expressions. Let "D" be Dana's current age and "M" be her mother's current age. Since we know "Dana is 20 years younger than her mother", we write: (1) D = M - 20Since we know "In 4 years, the mother will be twice as old as Dana", we write: (2) M + 4 = 2D + 8STEP 2:Next, we substitute D from the first equation into the second equation. M + 4 = 2D + 8M + 4 = 2(M - 20) + 8STEP 3:Finally, we solve forM. M + 4 = 2(M - 20) + 8M + 4 = 2M - 40 + 8M + 4 = 2M - 324 = M - 32M = 36The mother is 36 years old, so the correct answer ischoice B.

A 200-gallon tank has a pipe at the top that fills it with water at a rate of 15 gallons per minute. However, it also has a leak at the bottom that drains water at a rate of 6 gallons per minute. Assuming the tank is initially empty, how long would it take to fill the tank to 80% of its capacity? A. 14.4 minutes B. 15.3 minutes C. 17.8 minutes D. 19.2 minutes E. 20.2 minuteS

C (STEP 1: First, we calculate the rate at which the tank is filled, keeping in mind that water is both filled at 15 gallons per minute, and drained at 6 gallons per minute: STEP 2: Next, we calculate how much of the tank needs to be filled, keeping in mind that the 200-gallon tank needs to be filled to 80%: STEP 3: Finally, to calculate the time t it takes for the tank to be filled, we can write the following expression and solve for t)

Which of the following lines is perpendicular to the x-axis? A. y + 3x - 2 = 0 B. y = 5x C. x = -4 D. y = 4 E. y = x

C (Option A: y + 3x - 2 = 0 → y = -3x + 2When written in slope-intercept form (shown above), the slope is -3, so this option is incorrect. Option B: y = 5xThe slope is 5, so this option is incorrect. Option C: x = -4The slope here is infinity and -4 is a constant, so this line is perpendicular to the x-axis. Option D: y = 4This is the equation of a line parallel to the x-axis. This option is incorrect. Option E: y = xThis is the equation of a line with the slope of 1. This option is incorrect.)

Albert signed a contract to work 30 days under the following conditions: for each day he worked he was to receive 50 dollars. For each day he did not work he was to forfeit 15 dollars. At the end of 30 days, Albert received 980 dollars. How many days was Albert not working? A. 5 days B. 6 days C. 8 days D. 12 days E. 14 days

C. 8 days This problem tests your ability to solve real-world problems. There are two ways to solve this. Method #1- Algebraic Expressions This approach requires us to translate the information into algebraic expressions then solve use substitution. Let x be the number of days Albert worked and y be the number of days Albert didn't work. STEP 1: To start, we write the algebraic expressions. (1) x + y = 30(2) 50x - 15y = 980STEP 2:We then isolatexin the first expression. x + y = 30x = 30 - ySTEP 3:Next, we substitute the result fromstep 2forxin the second expression. 50x - 15y = 98050(30 - y) - 15y = 980STEP 4:Finally, we solve for the number of days Albert did not work ory. 50(30 - y) - 15y = 9801500 - 50y - 15y = 9801500 - 65y = 980-65y = -520y =-520/-65y = 8 daysThis means Albert didn't work for8out of the30days. Method #2 - Divide the Differences This is a shortcut we can use to get our answer. To do this, we will determine the total money Albert lost from the days he didn't work, then divide it by the amount of money Albert lost per day. STEP 1: First, we determine the total money Albert lost from the days he didn't work. $ Albert would have earned if he worked 30 days:50 x 30 = $1500Actual $ Albert earned:$980Total $ Albert lost from the days he didn't work:$1500 - $980 = $520STEP 2:Everyday Albert didn't work, he lost$50plus the$15that he forfeits. This totals to$65lost per day. STEP 3: Finally, we divide the total money lost by the money lost per day. 520/65= 8 daysThis gives the same answer asmethod #1. Therefore, the correct answer is choice C.

If 2x2 - ax + 6 = 0 has two distinct real roots, which of the following must be true? A. a2 < 48 B. a2 = 48 C. a2 > 48 D. a2 = 12 E. a2 = 24

C. a2 > 48 To solve this question, we must be familiar with the discriminant of a quadratic equation. Recall the general quadratic formula: ax2+ bx + c = 0Recall the discriminant formula: b2- 4acThe type of roots a quadratic equation has depends on the value of the discriminant: If b22 - 4ac < 0, the quadratic equation has no real roots. If b22 - 4ac = 0, the quadratic equation has two equal real roots (one root). If b22 - 4ac > 0, the quadratic equation has two distinct real roots. With this in mind, we will solve for the discriminant and then determine its relation to zero to get our answer. STEP 1: For our problem, the discriminant is: Quadratic ax2+ bx + c = 02x2- ax + 6 = 0a = 2, b = -a, c = 6Discriminant b2- 4ac(-a)2- 4(2)(6)a2- 48STEP 2:Since our quadratic equation has2 real roots, the discriminant is greater than zero. a2- 48 > 0STEP 3:We isolatea2by adding48to both sides of the equation. a2- 48 > 0a2> 48Therefore, the correct answer is choiceC. Video Solution

If the price of a concert ticket increases by 20%, then increases by another 20%, and then increases again by 10%, what is approximately the overall percent increase of the ticket? A. 40% B. 48.5% C. 50% D. 58% E. 60%

D (STEP 1: To start, we choose a starting price for the ticket that will make calculations easy. We choose 100. STEP 2: We determine the price of the ticket after the first 20% increase. Recall that we convert a percent into a decimal by placing a decimal two places to the left of the percent. 100 + 20% of 100= 100 + (0.2)100= 100 + 20= 120The price after the first20%increase is$120. STEP 3: Then, we determine the price of the ticket after the second 20% increase. We use the result from step 2 in our calculations. 120 + 20% of 120120 + (0.2)120120 + 24= 144The price after the second20%increase is$144. STEP 4: Now, we determine the price of the ticket after the last 10% increase. We use the result from step 3 in our calculations. 144 + 10% of 144144 + (0.1)144144 + 14.4= 158.40The final price after all three percent increases is$158.4. STEP 5: Finally, we find the overall percent increase. Since we chose 100 as the initial value, we can clearly see there is an increase of 58.4, or 58.4%.)

Dana's test scores on the first four exams are 80, 78, 70, and 60. If Dana wants to increase her test average by 8 points, what is the minimum average she needs on her last 2 exams to accomplish her goal? A. 98.5 B. 97.5 C. 96.8 D. 96.0 E. 94.0

D (STEP 1: To begin, we find the average of the first four tests. STEP 2: We then determine the desired average. Since the average of the first four tests is 72 and Dana wants to increase it by 8 points, we find the desired average to be 80 points for all 6 tests. STEP 3: We now find the combined score Dana needs on the last two tests. Let x be the combined score of the last two tests. Using the average formula, we write: This means Dana needs a combined score of at least 192 on her last two tests to get an average of 80% on all 6 tests. STEP 4: Finally, we divide 192 by 2 to find the average of the last two tests)

In how many ways can 3 different algebra books and 3 different physics books be arranged on a shelf? A. 9 B. 20 C. 90 D. 720 E. 780

D (This problem asks us to find the number of ways 6 total books can be arranged on a shelf. To solve this, we recall the following formula: n! = n(n - 1)(n - 2) ⋯ x 1 Where n is the total number of itemsIt's important to note when you substitute a number forninto the formula above, it is the same as the product of all decreasing integers from n to 1. For example, ifn = 4: n! = n(n - 1)(n - 2) ⋯ x 1 4! = 4(4 - 1)(4 - 2)(4 - 3) 4! = 4 x 3 x 2 x 1 With this in mind, we don't need to memorize the formula. Instead, we can use the shortcut to solve the problem. STEP 1: We first determine the value of n. Since there are 6 total books, n = 6. STEP 2: Then, we find the number of ways the 6 books can be arranged. 6! = 6 x 5 x 4 x 3 x 2 x 16! = 720 ways)

For a company that manufactures air conditioners, the fixed costs for the company amount to $60,000 regardless of how many air conditioners they sell. If the company produces air conditioners for $49 a unit and sells them for $99 a unit, what is the least amount of air conditioners the company must sell to break even? A. 750 B. 908 C. 1,104 D. 1,200 E. 1,350

D (STEP 1: First, we can calculate the revenue, which is the income of the company. Since the company plans to sell x number of air conditioners for $99 a unit, we can write: STEP 2: Next, we can calculate the costs. Since the company has a fixed cost of $60,000 and has to spend $49 for each of the x number of air conditioners they produce, we can write: STEP 3: Finally, we use the formula relating revenue and cost at the beginning of the explanation to write the following expression and solve for x:)

x2 + x - 6 = 0 Quantity A= X Quantity B= 0 A. Quantity A is greater. B. Quantity B is greater. C. The two quantities are equal. D. The relationship cannot be determined from the information given.

D (STEP 1: To begin, we factor the quadratic equation. x2+ x - 6 = 0(x - 2)(x + 3) = 0 STEP 2:We then set each term equal to zero and solve forx. x - 2 = 0x = 2x + 3 = 0x = -3 STEP 3:Finally, we compare the two quantities to each other. Since quantity A can equal either -3or2, we cannot properly compare both quantities. If quantity Ais-3, then quantity Bis greater. If quantity Ais2, then quantity Bis greater. Since the relationship between quantity A and B cannot be determined, the correct answer is choice D.)

Dennis has $4.00 in nickels dimes and quarters. If he has four more quarters than nickels, and 3 times as many dimes as nickels, how many dimes does he have? A. 9 B. 10 C. 11 D. 15 E. 17

D (STEP 1: We start by converting the given information into three equations. To keep calculations simple, we work with cents instead of dollars. Let N be the number of nickels, D the number of dimes, and Q the number of quarters. When we do this, we get the following equations: 5N + 10D + 25Q = 400We can reduce this equation by5to get: (1) N + 2D + 5Q = 80(2) Q = 4 + N(3) D = 3NSince the question is asking about the number of dimes, we can see from the third equation that the answer must be a multiple of3. We cross out choicesB,C, andE. STEP 2: Next, we substitute equations 2 and 3 into equation 1 to determine the number of nickels. We do this because we need to find N before we can find D. N + 2D + 5Q = 80N + 2(3N) + 5(4 + N) = 80N + 6N + 20 + 5N = 8012N + 20 = 8012N = 60N =60/12= 5STEP 3:We then substitute5forNinequation 3to determine the number of dimes. D = 3 ND = 3 x 5 D = 15)

James drives from City A to City B at a speed of 70 mph. Three hours later, Grant also starts driving from City A to City B, but at a speed of 100 mph. If they continue driving at their respective speeds, how many miles away from City A will they meet? A. 443 B. 550 C. 675 D. 700 E. 815

D (WE will need the following formula that relates to distance, speed, and time: STEP 1: Because James and Grant will meet at the same distance away from City A, we can write: STEP 2: Using the distance, speed, and time formula, we can now rewrite both James and Grant's distances using speed and time: STEP 3: Now, we substitute James and Grant's driving speed from the question statement and set t as the time it takes for Grant to drive, and t + 3 as the time it takes for James to drive (because James departs 3 hours earlier than Grant, so we add three hours to his driving time): STEP 4: Now, we solve for t, which is the time it takes for Grant to drive: STEP 5: Finally, because the question asks for the distance from City A in which they meet, we simply use the distance-speed-time formula to find the distance driven)

In a cookie bag, the ratio of almond cookies to raisin cookies to peanut butter cookies is 4:3:7. If there are 280 cookies in the bag, how many raisin cookies are there? A. 25 B. 30 C. 45 D. 50 E. 60

E (STEP 1: To begin, we write an equation that shows the relationship between all cookies. Let x be the constant of proportionality. Since we know there is a 4:3:7 ratio and a total of 280 cookies, we write: STEP 2: We then solve for x. STEP 3: Finally, we multiply the ratio of raisin cookies by the constant of proportionality to determine the number of raisin cookies. Since the ratio of raisin cookies is 3, we write: There are 60 raisin cookies so the correct answer is choice E. Note: To check your work, simply repeat step 3 for almond and peanut butter cookies and take the sum of all three cookie types. If they add up to 280, you are 100% right)

In a summer camp of 220 students, 150 have blond hair, 80 are left-handed, and 30 have blond hair and are left-handed. How many students in the camp are neither blond nor left-handed? A. 8 B. 10 C. 13 D. 15 E. 20

E (STEP 1: We set up our information as a Venn diagram. Since there are 30 students that are blonde and left-handed, we write: STEP 2: We find the number of blonde students only and the number of left-handed students only. Blonde only = 150 - 30 = 120 Left-handed only = 80 - 30 = 50 STEP 3: Finally, we determine the number of students that are neither blonde nor left-handed. To do this, we recognize the number of students that are only blonde, only left-handed, or both is 120 + 30 + 50 = 200. Since we know the total number of students is 220, the remaining 20 students are neither blonde nor left-handed)

Lily and Nada share silver coins in the ratio 7:3. If Lily gives 3 silver coins to Nada, the ratio becomes 5:3. How many silver coins did Lily have initially? A. 14 B. 22 C. 28 D. 30 E. 36

c (this is another variation of a ratio problem that is commonly tested on the OAT. In order to solve this, we need to set up a proportion to represent the ratios. We then solve for the common factor and use the result to get our answer. STEP 1: To begin, we write a proportion that shows the relationship between the initial and new ratios. Let 7x and 3x be the initial amount of silver coins Lily and Nada had, respectively. If Lily gives 3 coins to Nada, Lily will then have 7x - 3 coins and Nada will have 3x + 3 coins. We show this below: STEP 2: We then cross multiply to solve for x. STEP 3: Finally, we determine the number of coins Lily initially had by plugging in 4 for x in 7x.)