#18 Data Analysis, Interpretation, and Sufficiency - Probability, Statistics, Applied Mathematics
Josh went for a 10-mile run. How many dogs did he see during his run? 1) Josh saw 10 less dogs than cats2) Josh saw 13 cats
Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient Statement (1) tells us Josh saw 10 fewer dogs than cats. However, without knowing how many cats Josh saw, statement (1) is insufficient. Statement (2) tells us Josh saw 13 cats. This is the information that we wanted when considering statement (1)! Combining both statements, we know that Josh saw 13 cats and 13 - 10 = 3 dogs. Therefore, both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. Answer choice [C] is correct.
Tim has only dimes. Lucas has only quarters. Who has more money? 1) Tim has 10 more dimes than Lucas has quarters2) Tim has 20 dimes
Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient Statement (1) tells us that Tim has 10 more dimes than Lucas has quarters. This alone does not tell us who has more money, but let's keep it in mind when considering statement (2). Statement (2) tells us Tim has 20 dimes. With statement (1), we know that Lucas has 10 quarters. So, Tim has $2.00 and Lucas has $2.50. Therefore, Lucas has more money. Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. Answer choice [C] is correct.
Tyler has a box of pins and needles. How many needles does Tyler have? 1) Tyler has three pins2) Tyler has twice as many needles as pins
Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient Statement (1) tells us that Tyler has three pins. However, we can't determine how many needles he has, so statement (1) alone is insufficient. Statement (2) tells us Tyler has twice as many needles as pins. We'd need to know how many pins Tyler has to determine how many needles he has, so statement (2) alone is also insufficient. If we consider both statements, then we know that Tyler has 3 pins and 6 needles. Therefore, both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. Answer choice [C] is correct.
Jennie owns five convertibles, at least one of which is red. What is the probability of blindly choosing a red convertible? 1) There is a 1/5 chance of choosing a green convertible2) There is a 3/5 chance of choosing a blue, yellow, or silver convertible
Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient The probability of an event occurring is the number of outcomes in which the event occurs, divided by the total number of equally likely outcomes. We know Jennie has five convertibles, so to find the probability of choosing a red convertible, we need to know how many of Jennie's cars are red. Statement (1) tells us the probability of choosing a green convertible is 1/5. Since Jennie has 5 convertibles, this means that exactly one of her cars is green. However, we don't know how many are red, so statement (1) alone is insufficient. Statement (2) tells us the probability of choosing a blue, yellow, or silver convertible is 3/5. This means that Jennie has 3 non-red cars that are either blue, yellow, or silver. We need to combine this information with statement (1): with both statements, we know that Jennie has 4 cars that are not red. Since she has at least 1 red car, her fifth car must be red and the probability of choosing a red convertible is 1/5. Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. Answer choice [C] is correct.
Josh made sandwiches. How many did he make? 1) Josh worked for one hour and 40 minutes2) Josh worked at a rate of four sandwiches per 10 minutes
Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient This is a distance = rate × time type of question. We have a basic relation: the number of sandwiches is equal to the rate of sandwich-making multiplied by time spent making sandwiches. Statement (1) tells us the time that Josh spent making sandwiches. Statement (2) tells us the rate at which Josh can make sandwiches. Both statements together let us solve for the number of sandwiches Josh made. Therefore, both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. Answer choice [C] is correct.
A student's daily routine consists of travel from home to school and from school to home. The student travels to school in 30 minutes. What is the student's average speed for the daily routine? 1) The student travels at the same speed going to and from school 2) The school is two miles away from the student's home
Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient To determine the student's average speed, we must determine the total distance traveled (to and from school) and the total time needed for the round trip (we cannot assume that the student's speed is constant). Statement (1) tells us that the student travels at the same speed going to and from school. However, we have no information about the distance from home to school, so statement (1) alone is insufficient. Statement (2) tells us that school is two miles away from the student's home, so the total distance for a round trip is 2 × 2 = 4 miles. However, since we don't know if the student traveled at a constant speed the entire trip, statement (2) alone is also insufficient. If we combine the two statements, we know that the student's speed was constant, so it takes the student one hour to make a round trip. Since the total distance traveled is 4 miles, the student's average speed is 4 miles ÷ 1 hour = 4 mi/hr. Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. Answer choice [C] is correct.
Points A, B, C, and D are on a number line, not necessarily in that order. Point C is between A and D. The distance between A and D is twice the distance between C and D. Point B is to the left of point A. If the distance between B and D is 37 and the distance between B and C is 22, what is the distance between A and B? 1) The distance between A and C is 15 2) The distance between C and D is 15
EACH statement ALONE is sufficient
Amy is six years younger than Brandon. How old is Brandon now? 1) Eight years ago, Brandon was twice as old as Amy2) In two years, Amy will be 17 years old
EACH statement ALONE is sufficient Let b represent Brandon's current age, and let a represent Amy's current age. From the question prompt, we know b = a + 6. We would just need another independent equation involving a and b to solve for Brandon's age. Statement (1) tells us that 8 years ago, Brandon was twice as old as Amy. We can translate this into math: b - 8 = 2(a - 8) b = 2a - 8 This equation is independent of our first equation, so statement (1) alone is sufficient. Statement (2) tells us that in 2 years, Amy will be 17 years old. Amy is currently 15 years old, so Brandon must be 15 + 6 = 21 years old. Statement (2) alone is sufficient. EACH statement ALONE is sufficient. Answer choice [D] is correct.
If two sodas and one candy bar cost $5, how much does one soda and two candy bars cost? 1) A soda costs $1 more than a candy bar 2) Two candy bars are the same cost as one soda
EACH statement ALONE is sufficient Let s be the cost of a soda, and let c be the cost of a candy bar. Since 2 sodas and 1 candy bar cost $5, we have the equation 2s + c = 5. We would need another unique, independent equation involving s and c in order to solve for their values and determine the value of s + 2c. Statement (1) tells us that a soda costs $1 more than a candy bar, so s = c + 1. This equation is independent of our first equation, so we have enough information to find the cost of 1 soda and 2 candy bars. Statement (1) alone is sufficient. Statement (2) tells us that 2 candy bars are the same cost as 1 soda, so 2c = s. This equation is also independent of our first equation, so we have enough information to find the cost of 1 soda and 2 candy bars. Statement (2) alone is sufficient. Therefore, EACH statement ALONE is sufficient. Answer choice [D] is correct.
Does circle C or square S have a larger area? 1) Square S can fit inside of circle C2) The radius of circle C is larger than a side length of square S
EACH statement ALONE is sufficient Statement (1) tells us that Square S can fit inside Circle C, which means that the area of Square S must be smaller than the area of Circle C. Statement (1) alone is sufficient. Statement (2) tells us that the radius of Circle C is larger than the side length of Square S. The area of a circle is equal to 𝜋r2, while the area of a square is equal to s2. Squaring a larger positive number will result in a larger value than squaring a smaller positive number, so the area of Circle C must be greater than the area of Square S. Statement (2) alone is sufficient. EACH statement ALONE is sufficient. Answer choice [D] is correct.
A drawer contains ten marbles, colored red, green, and blue. What is the probability of randomly selecting a green marble? 1) There are three red marbles and five blue marbles 2) There are two green marbles
EACH statement ALONE is sufficient The question is asking us for the probability of an event occurring. The probability of an event occurring is the number of ways the desired outcome can occur, divided by the total number of possible outcomes. Here, there are 10 possible outcomes, since there are 10 different marbles we could pull from the drawer. To determine the probability of picking a green marble, we need to know how many green marbles are in the drawer. Statement (1) tells us there are 3 red marbles and 5 blue marbles in the drawer. Therefore, there are 10 - 3 - 5 = 2 green marbles, and the probability of picking a green marble is 2/10. Statement (1) alone is sufficient. Statement (2) tells us explicitly there are 2 green marbles, so the probability of picking a green marble is just 2/10. Statement (2) alone is also sufficient. Therefore, EACH statement ALONE is sufficient. Answer choice [D] is correct.
Haley leaves home at 10:30 a.m. Haley drives 40 miles from her home to Point A. She then drives from Point A to Point B. Finally, Haley drives 120 miles from Point B to home. She returns home at 5:30 p.m. What is Haley's average speed over her entire trip? 1) The distance from Point A to Point B is 40 miles 2) The distance from Point A to Point B is fifty miles
EACH statement ALONE is sufficient. Because Haley left home at 10:30 a.m. and returned at 5:30 p.m., her total travel time is 7 hours. In order to determine her average speed, we must also know the total distance traveled. Haley drives 40 miles from home to Point A, and 120 miles from Point B to home. However, we need to know the distance between Point A and Point B in order to calculate the total distance Haley drove. Statement (1) tells us that the distance from Point A to Point B is 40 miles, so statement (1) alone is sufficient. Statement (2) tells us that the distance from Point A to Point B is 50 miles, statement (2) alone is sufficient.
A hiker wants to buy a tent from a local store. The original price for a tent is $553. The hiker only has $440. Can she buy a tent? 1) The discounted price for a tent is $100 more than half of the original price 2) The discounted price for a tent is $100 more than one-third of the original price
EACH statement ALONE is sufficient. The original price of a tent is $553, but the hiker only has $440. To determine if the hiker can afford the tent, we need to know the discounted price of the tent. Statement (1) tells us that the discounted price is $100 more than half the original price. Since we know the original price, we can solve for the discounted price. Statement (1) alone is sufficient. Statement (2) tells us that the discounted price is $100 more than one-third of the original price. Similarly, we can solve for the discounted price because we know the original price. Statement (2) alone is sufficient.
A computer is being sold at a discount for $350. Was the original list price greater than $400? 1) The discounted price is $50 less than the original price 2) The discounted price is $50 more than one-third of the original price
EACH statement ALONE is sufficient. We are given that the computer's price after a discount is $350 and want to know if the original price was greater than $400. Let the variable P represent the original price. Statement (1) says that the discounted price is $50 less than the original price. P - 50 = 350 P = $400 The original price was not greater than $400, so statement (1) alone is sufficient. Statement (2) tells us the discounted price is $50 more than one-third of the original price. 50 + (1/3)P = 350 P = $900 The original price was greater than $400, so statement (2) alone is sufficient.
If Pete has $0.12 in his pocket, how many coins (pennies = $0.01, nickels = $0.05, dimes = $0.10, quarters = $0.25) does he have? 1) Pete has one dime 2) Pete has no nickels
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (1) tells us Pete had 1 dime, which is $0.10. That leaves $0.02 unaccounted for, so we know Pete must also have 2 pennies, for a total of 3 coins. Statement (1) alone is sufficient. Statement (2) tells us Pete has no nickels. However, Pete could have 1 dime and 2 pennies, or 12 pennies. Therefore, we can't determine how many coins he has from statement (2) alone. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Answer choice [A] is correct.
Amber has a standard 52-card deck. She chooses two cards with replacement. Did she choose two aces in a row? 1) The first card chosen was a jack.2) The probability of drawing a specific card is less than 2%
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (1) tells us that the first card Amber chose was a jack, so Amber did not choose two aces in a row. Statement (1) alone is sufficient. Statement (2) tells us the probability of drawing a specific card is less than 2%. This actually isn't new information and doesn't tell us anything about the cards Amber chose. There are 52 unique cards in the deck, and so the probability of choosing any specific card is 1/52. Statement (2) alone is insufficient.
The chart below shows the snow total in inches for four different ski resorts from Monday through Friday. If one resort saw more snow than any other on Saturday, which resort was that? 1) From Monday through Saturday, each resort had the most snow, or tied for the most snow, at least one day. 2) From Monday through Saturday, no resort had the most snow, or tied for the most snow, every day.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement 1 tells us that each resort had the most snow (or tied for most snow) on at least one day on Monday through Saturday. Since the graph shows that Eagle Peak, Cornice City, and Powder Mountain had the most snow on Monday through Friday, Chute and Sweet must have had the most snow on Saturday. Thus, Statement 1 alone is sufficient. Statement 2 tells us that no resort had the most snow on every day between Monday through Saturday, but we know this from the graph of Monday through Friday. Thus, this statement offers no new useful information and Statement 2 alone is insufficient. Therefore, answer choice [A] is correct.
What is the mean of p, q, and r? 1) p + q + r = 15 2) p = 15
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. The mean of p, q, and r is equal to the sum of all the terms (p + q + r) divided by the number of terms (3). Statement (1) tells us that p + q + r = 15. The mean is equal to 15 ÷ 3 = 5, so statement (1) alone is sufficient. Statement (2) tells us that p = 15. However, because we don't know the values of the other two variables, we can't determine the mean. Statement (2) alone is insufficient. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Answer choice [A] is correct.
A coin is flipped n times. Is the probability of flipping all tails less than 1/16? n is greater than 0. 1) n = 4 2) n > 3
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. This question asks us about successive probabilities. Recall that the probability of an event occurring is the number of outcomes in which the event occurs, divided by the total number of equally likely outcomes. In the case of a coin flip, there are 2 equally likely outcomes: heads or tails. There is only 1 desired outcome (tails), so the probability of flipping a coin once and getting tails is 1/2. To determine the probability of getting two consecutive tails, we multiply the probabilities: (1/2) × (1/2) = (1/2)^2. In general, if we flip a coin n times, the probability of getting all tails is (1/2)^n. Statement (1) tells us that n = 4. The probability of flipping 4 consecutive tails is (1/2)^4 = 1/16. Statement (1) alone is sufficient to determine that the probability is not less than 1/16. Statement (2) tells us that n > 3, so n is a number 4 or larger. We already saw that if n = 4, then the probability of flipping 4 tails is not less than 1/16. However, for any n larger than 4, (1/2)^n will be smaller than 1/16. For example, (1/2)^6 = 1/32 < 1/16. Therefore, statement (2) alone is not sufficient. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Answer choice [A] is correct.
A wallet contains an equal number of pennies, nickels, and dimes. There are no quarters in the wallet. How many of each type of coin does the wallet contain? 1) In total, the coins equal $1.12 2) The number of coins in the wallet is greater than twenty
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. We know that there are an equal number of pennies, nickels, and dimes. Let this number be x; we want to solve for x. Statement (1) tells us that the total worth of all the pennies, nickels, and dimes is 112 cents. The worth of x pennies is x cents, the worth of x nickels is 5x cents, and the worth of x dimes is 10x cents. We can write the equation x + 5x + 10x = 112 and solve for x. Statement (1) alone is sufficient. Statement (2) tells us that the total number of coins is greater than 20. The total number of coins is 3x, so we can write 3x > 20. However, this does NOT provide us with enough information to solve for x. Statement (2) alone is insufficient.
Is the median of set A greater than the mean? 1) The mode of set A is equal to the median2) All the values in set A are the same
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. Let's start by thinking through each statement separately. Statement (1) says that the median (middle value) is equal to the mode (most commonly occurring value). However, it says nothing about the other values in the set, so we cannot determine anything about the mean. For example, the two sets {49, 50, 50, 50, 100} and {1, 50, 50, 50, 51} both have an equal median and mode. However, the mean of the first set is higher than the median, and the mean of the second set is lower than the median. Statement (1) alone is insufficient. Statement (2) says that all values in set A are the same; this value will be the mode, the mean, and the median. Therefore, we know that median is not greater than the mean, and statement (2) alone is sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. Answer choice [B] is correct.
What is Alex's average speed in miles per hour over two days? 1) Alex traveled five miles today and 10 miles yesterday 2) Alex traveled for 3 hours today
Statements (1) and (2) TOGETHER are NOT sufficient In order to determine Alex's average speed over two days, we need to know the total distance he traveled and the total time spent traveling over these two days. Statement (1) tells us that Alex traveled 5 miles today and 10 miles yesterday. However, we do not know the time it took Alex to travel these distances, so statement (1) alone is insufficient. Statement (2) tells us that Alex traveled for 3 hours today. We have no information about the distance traveled on either day or the time he spent traveling yesterday. Statement (2) alone is insufficient. If we combine the two statements, we know that Alex traveled 5 miles in 3 hours today. However, we still don't know the time it took for Alex to travel 10 miles yesterday.
Is Amber older than Samantha? 1) Amber was born in a leap year 2) Amber and Samantha were born the same year
Statements (1) and (2) TOGETHER are NOT sufficient Statement (1) is not sufficient on its own because Amber being born in a leap year tells us nothing about her age relative to Samantha's age. Statement (2) is not sufficient on its own either. It only tells us that Amber's age and Samantha's age are within one year of each other. Even when both statements are combined, we have no clear indication of who is older. Therefore, Statements (1) and (2) TOGETHER are NOT sufficient. Answer choice [E] is correct.
A car salesman bought three used cars for $33,000 each. After spending an additional $4,000 to fix each car, he wants to make a $7,500 profit in total. After successfully selling all three cars, how much profit did he make? 1) Car 1 sold for $7,500 more than the price it was bought at 2) Car 2 sold for $50,000
Statements (1) and (2) TOGETHER are NOT sufficient To find the total profit the salesman makes from selling all 3 cars, we need to know the profit he makes on each car. Statement (1) tells us he sold Car 1 for $7,500 for more than the price it was bought at, so he made a profit of $3,500 ($7,500 - $4,000) from selling Car 1. Statement (2) tells us that Car 2 was sold for $50,000, so his profit was $50,000 - ($33,000 + $4,000). DAT Pro-tip: Note that in this problem, the statement "he wants to make a $7,500 profit in total" didn't give us any useful information. It helps to keep in mind what exactly the problem is asking (here, the total profit he actually makes) in order to discern if a certain piece of information is relevant or not.
Aaron and Cindy ran a race together. What is Cindy's average speed for the race? 1) Aaron started the race 15 minutes before Cindy 2) The race is three miles long
Statements (1) and (2) TOGETHER are NOT sufficient. In order to determine Cindy's average speed for the race, we need to know the distance she ran and the time it took for her to complete the race. Statement (1) tells us that Aaron started the race 15 minutes before Cindy. This doesn't tell us anything about the time it took either person to complete the race, so statement (1) is insufficient. Statement (2) tells us that the race was three miles long. While we know the distance, we still don't know Cindy's time. Statement (2) is insufficient. Because statement (1) doesn't tell us about Cindy's time, combining the statements is also not sufficient to determine her average speed. Therefore, statements (1) and (2) TOGETHER are NOT sufficient. Answer choice [E] is correct.
There are 33 marbles in a bag, at least one of which is green. How many are green? 1) There are three marble colors in the bag 2) 14 marbles are orange
Statements (1) and (2) TOGETHER are NOT sufficient. Statement (1) tells us that there are 3 different colors of marbles in the bag. Alone, this is insufficient, because it doesn't tell us anything about how many marbles there are of each color. Statement (2) tells us there are 14 orange marbles. Alone, this does not tell us anything about how many green marbles there are. If we combine both statements, we know that there are 3 different colors and 14 out of 33 marbles are orange. The number of marbles that are green and the number of marbles that are the third color must add up to 19. Since there is at least 1 green marble, there could be any number of green marbles from 1 to 18. We cannot determine the exact number of green marbles. Statements (1) and (2) TOGETHER are NOT sufficient. Answer choice [E] is correct.
