Test: Distance/Rate Problems - GMAT MCQ

From statement (1), we know that Danielle started the race at 8:00 a.m. However, we don't have any information about her speed or the duration of the race. Therefore, statement (1) alone is not sufficient to determine the time she finished.

From statement (2), we know that at 9:30 a.m., Danielle was halfway through the race, and at 10:00 a.m., she was 2/3 of the way through the race. This implies that the second half of the race took 30 minutes (from 9:30 a.m. to 10:00 a.m.), and the last third of the race also took 30 minutes (from 9:30 a.m. to 10:00 a.m.). However, we still don't have information about Danielle's speed or the length of the entire race. Therefore, statement (2) alone is not sufficient to determine the time she finished.

By combining both statements, we know that Danielle started the race at 8:00 a.m., and at 10:00 a.m., she was 2/3 of the way through the race. This means she took 2 hours (from 8:00 a.m. to 10:00 a.m.) to complete 2/3 of the race. However, we still don't have enough information to determine the exact time she finished or the speed at which she ran.

Therefore, both statements together are sufficient to determine the time Danielle finished. The answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

From statement (1), we know that Wendy drove 15 miles in 20 minutes. However, this statement alone does not provide any information about Marty's distance.

From statement (2), we know that Marty drove at the same average speed as Wendy. However, we don't have any information about Wendy's average speed, so we cannot determine Marty's distance based on this statement alone.

Since neither statement alone is sufficient to determine Marty's distance, and we don't have any information that allows us to combine the statements and calculate the distance, the answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

To determine whether Gurjit managed to reach back home latest by 6:50 am, let's analyze each statement:

Statement (1): Gurjit reached the pole at exactly 6:20 am.
This statement tells us the time it took Gurjit to reach the pole. Since he started running at 6:00 am and reached the pole at 6:20 am, we know that it took him 20 minutes to reach the pole. However, this information alone is not sufficient to determine if he managed to reach back home by 6:50 am.

Statement (2): Gurjit's average speed while running back home from the pole was 20% lesser than his average speed while running towards the pole.
This statement provides information about Gurjit's speed while returning from the pole. However, without knowing the actual speeds or the distance between the pole and Gurjit's home, we cannot determine whether he managed to reach back home by 6:50 am based on this statement alone.

Considering both statements together, we know that Gurjit took 20 minutes to reach the pole. However, since we don't have information about his speed while running towards the pole or the distance between the pole and his home, we still cannot determine if he reached back home by 6:50 am.

Therefore, both statements together are not sufficient to answer the question asked. The correct answer is (C): BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): A and B both start from the same point and cover the same distance before their first meeting.
This statement indicates that A and B meet at the same point on the track after covering the same distance. However, it does not provide any information about the time it took for them to meet or the speeds at which they were running. Without any information about the time or speed, we cannot determine the sum of their respective speeds. Thus, statement (1) alone is not sufficient to answer the question.

Statement (2): A can cover the entire circular track in 10 minutes.
This statement provides information about A's time to complete one full lap around the circular track. However, it does not provide any information about B's speed or their relative positions when they start. Without any information about B's speed, we cannot determine the sum of their respective speeds. Hence, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we know that A and B meet at the same point after covering the same distance, but we still lack information about their speeds or the time it took for them to meet. Therefore, even when considering both statements, we cannot determine the sum of their respective speeds. The correct answer is (C): BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): Runner A ran onto the track 10 seconds after Runner B ran onto the track and ran off the track 8 seconds after Runner B ran off the track.
This statement provides information about the time difference between Runner A and Runner B entering and exiting the track. However, it does not give any information about the actual speeds at which they were running. Without the speed information, we cannot determine the time it took for Runner A to run the track. Thus, statement (1) alone is not sufficient to answer the question.

Statement (2): Runner B ran the track at a constant speed of 9 miles per hour.
This statement provides the speed at which Runner B ran the track, but it does not provide any information about Runner A's speed or the time difference between their runs. Without Runner A's speed or the time difference, we cannot determine how long it took Runner A to run the track. Therefore, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we know the time difference between Runner A and Runner B entering and exiting the track (from statement 1) and the speed at which Runner B ran the track (from statement 2). However, we still lack information about Runner A's speed. Without Runner A's speed, we cannot determine the time it took for Runner A to run the track. Therefore, even when considering both statements together, we cannot answer the question. The correct answer is (C): BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): Officer Johnson noted that the driver had traveled 30 miles from point A to point B on I-75.
This statement provides information about the distance traveled by the driver, but it does not give any details about the time it took to cover that distance or the driver's speed. Without information about the time or speed, we cannot determine whether the driver exceeded the speed limit. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): Officer Johnson noted that it took the driver 30 minutes to travel from point A to point B on I-75.
This statement provides information about the time it took for the driver to cover the distance between points A and B, but it does not give any information about the actual distance or the driver's speed. Without the distance or speed information, we cannot determine whether the driver exceeded the speed limit. Hence, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we know the distance traveled by the driver (from statement 1) and the time it took to travel that distance (from statement 2). By calculating the speed as distance divided by time, we can determine whether the driver exceeded the speed limit:

Speed = Distance / Time = 30 miles / 30 minutes = 1 mile per minute

Since the speed is given in miles per minute, we cannot directly compare it to the 50 mile-per-hour speed limit. We would need to convert the speed to miles per hour to make a valid comparison. Without additional information to convert the speed, we still cannot determine whether the driver exceeded the speed limit. Therefore, even when considering both statements together, we cannot answer the question. The correct answer is (C): BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): It took 3 minutes for the bicycle wheel to travel the entire distance.
This statement provides information about the time it took for the wheel to cover the distance, but it does not give any details about the wheel's rotation or the speed at which it was rotating. Without information about the rotation, we cannot determine the number of 360-degree rotations the wheel made. Hence, statement (1) alone is not sufficient to answer the question.

Statement (2): The wheel made twenty 360-degree rotations per minute.
This statement provides information about the number of 360-degree rotations the wheel made per minute, but it does not give any information about the distance traveled or the time it took to travel that distance. Without information about the distance or time, we cannot determine the number of rotations the wheel made while rolling 150 meters. Therefore, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we know the time it took for the wheel to travel the entire distance (from statement 1) and the number of 360-degree rotations the wheel made per minute (from statement 2). With this information, we can determine the number of rotations the wheel made while rolling 150 meters. By converting the time given in statement 1 from minutes to seconds (3 minutes = 180 seconds) and multiplying it by the rotations per minute given in statement 2 (20 rotations/minute), we can calculate the total rotations:

Total rotations = (Rotations per minute) × (Time in minutes) = 20 rotations/minute × 3 minutes = 60 rotations

Therefore, both statements together are sufficient to answer the question. The correct answer is (C): BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): Karen flew 3,000 miles.
This statement provides information about the distance Karen flew but does not give any details about the duration or the actual speed at any point during the flight. Without any information about time or speed, we cannot determine whether Karen exceeded 650 miles per hour. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): Karen flew for 5 hours.
This statement provides information about the duration of Karen's flight but does not give any details about the distance or the actual speed at any point during the flight. Without any information about distance or speed, we cannot determine whether Karen exceeded 650 miles per hour. Hence, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we still lack information about the speed at any point during the flight. Without the specific speed data, we cannot determine whether Karen ever exceeded 650 miles per hour. Therefore, the correct answer is (E): Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

To determine the time it took for the bus to travel from town A to town B, let's evaluate each statement separately:

Statement (1): The car's average speed for the trip was 5/8 of the average speed of the bus.
This statement alone provides a direct relationship between the car's speed and the bus's speed. Since the car took 2 hours to travel the distance, we can set up the equation:

Distance = Speed × Time

Let's denote the bus's average speed as B and the car's average speed as C. According to the statement, C = (5/8)B. We can rewrite the equation as:

Distance = C × Time

Since the distance between towns A and B is constant, we can equate the two equations:

C × 2 = B × Time

From this equation, we can solve for Time:

Time = (C × 2) / B = (5/8)B × 2 / B = 10/8 = 5/4

So, the bus took 5/4 hours, which is 1 hour and 15 minutes.

Statement (2): Towns A and B are 120 miles apart.
This statement alone provides the distance between towns A and B but does not give any information about the speeds or times of the car and the bus. Therefore, we cannot determine the time it took for the bus to travel the distance based on this statement alone.

Since we were able to determine the time taken by the bus using statement (1) alone, but statement (2) alone was insufficient, the correct answer is (A): Statement (1) alone is sufficient, but statement (2) alone is not sufficient to answer the question asked.

From statement (1), we know that Sam's average speed is 80% of Bill's average speed. However, we don't have any information about their actual speeds or how their speeds relate to the time taken. Therefore, statement (1) alone is not sufficient to determine how long it took Sam.

From statement (2), we know the distance from the school to the library is two miles. However, we don't have any information about the speeds or time taken by Bill and Sam. Therefore, statement (2) alone is not sufficient to determine how long it took Sam.

By combining both statements, we know that Sam's average speed is 80% of Bill's average speed, and the distance is two miles. Since both Bill and Sam traveled the exact same route, we can conclude that Sam took 80% of the time it took Bill. Since Bill took 12 minutes, Sam took 80% of 12 minutes, which is 9.6 minutes.

Therefore, both statements together are sufficient to determine how long it took Sam. The answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.