Test: Min/Max Problems - GMAT MCQ

Statement 1: The largest integer is twice the smallest integer.
This statement gives us a relationship between the largest and smallest integers, but it doesn't provide specific values. Without knowing the actual values of the integers, we cannot determine their individual values or the value of the middle integer. Therefore, statement 1 alone is not sufficient to determine the values of the integers.

Statement 2: One of them is 9.
This statement gives us the specific value of one of the integers, which is 9. However, it doesn't provide any information about the other two integers or their relationship. Therefore, statement 2 alone is not sufficient to determine the values of the integers.

When we consider both statements together, we can conclude the following:

• The largest integer is twice the smallest integer.
• One of the integers is 9.

From statement 2, we know that one of the integers is 9. If we consider this as the largest integer, then the smallest integer would be 9/2 = 4.5, which is not a positive integer. Therefore, the largest integer cannot be 9.

Since the integers are positive and the largest integer is twice the smallest integer, we can conclude that the smallest integer is 1, and the largest integer is 2. The middle integer is then (1 + 2 + 9)/3 = 4.

Therefore, the values of the integers are 1, 4, and 9.

The answer is option A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement 1: X < 8.
This statement tells us that the total number of pieces of equipment purchased, denoted by X, is less than 8. However, it doesn't provide any specific information about the distribution of beakers, flasks, and burners. Therefore, statement 1 alone is not sufficient to determine the number of flasks purchased.

Statement 2: Total cost of X pieces of equipment is $27.
This statement gives us information about the total cost of the equipment but doesn't provide any details about the individual quantities of beakers, flasks, and burners. Without knowing the specific quantities, we cannot determine the number of flasks purchased. Therefore, statement 2 alone is not sufficient to answer the question.

When we consider both statements together, we still don't have enough information to determine the number of flasks purchased. The statements provide limited information about the total number of pieces and the total cost but don't give any specific breakdown of quantities.

Therefore, the answer is option E: Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Statement 1: x + y + z = 67
This statement gives us the sum of the three integers but doesn't provide any information about the individual values or their positivity. However, we can determine the number of positive integers by considering the value of z. If z is positive, then at least one of x or y must also be positive. Therefore, statement 1 alone is sufficient to answer the question.

Statement 2: x + y = 40
This statement gives us the sum of two integers but doesn't provide any information about z or the individual positivity of x and y. Without knowing the value of z or having more information, we cannot determine the number of positive integers. Therefore, statement 2 alone is not sufficient to answer the question.

When we consider both statements together, we know the sum of all three integers (x + y + z = 67) and the sum of two integers (x + y = 40). However, we still don't have enough information to determine the number of positive integers. We need additional information about the values of x, y, and z to make a definitive conclusion.

Therefore, the answer is option A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement 1: w = max(20, z) for some integer z.
This statement tells us that the value of w is equal to the maximum of 20 and some integer z. However, it doesn't provide any information about the specific value of z or how it relates to the value of w. Therefore, statement 1 alone is not sufficient to determine the value of min(10, w).

Statement 2: w = max(10, w)
This statement tells us that the value of w is equal to the maximum of 10 and w itself. This implies that w must be greater than or equal to 10. However, it doesn't provide any specific value for w. Therefore, statement 2 alone is not sufficient to determine the value of min(10, w).

When we consider both statements together, we know that w is equal to the maximum of 10 and itself (w = max(10, w)). This means that w must be greater than or equal to 10. Additionally, statement 1 tells us that w is equal to the maximum of 20 and some integer z. Since 20 is greater than 10, we can conclude that w must be equal to 20.

Therefore, each statement alone is sufficient to determine the value of min(10, w), and the answer is option D: EACH statement ALONE is sufficient to answer the question asked.

Statement 1: One of the six countries sent 41 representatives to the congress.
This statement tells us that one of the countries sent 41 representatives, but it doesn't provide any information about the number of representatives sent by Country A. Therefore, statement 1 alone is not sufficient to determine whether Country A sent at least 10 representatives.

Statement 2: Country A sent fewer than 12 representatives to the congress.
This statement provides information specifically about Country A, stating that it sent fewer than 12 representatives. However, it doesn't give us the exact number of representatives sent by Country A. Therefore, statement 2 alone is not sufficient to determine whether Country A sent at least 10 representatives.

When we consider both statements together, we still don't have enough information to determine whether Country A sent at least 10 representatives. Statement 1 tells us that one of the countries sent 41 representatives, but we don't know if that country is Country A or if Country A sent more or fewer representatives than that. Statement 2 tells us that Country A sent fewer than 12 representatives, but we don't know the exact number.

Without additional information about the number of representatives sent by Country A or the relationship between Country A and the country that sent 41 representatives, we cannot determine whether Country A sent at least 10 representatives.

Therefore, the answer is option E: Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Statement 1: The farm has more than twice as many cows as pigs.
This statement provides a comparison between the number of cows and pigs on the farm, stating that there are more cows than twice the number of pigs. However, it doesn't give us specific information about the total number of cows or pigs. Therefore, statement 1 alone is not sufficient to determine the number of cows.

Statement 2: The farm has more than 12 pigs.
This statement tells us that there are more than 12 pigs on the farm. However, it doesn't provide any information about the number of cows or the ratio between cows and pigs. Therefore, statement 2 alone is not sufficient to determine the number of cows.

When we consider both statements together, we still don't have enough information to determine the exact number of cows. Statement 1 provides a relationship between the number of cows and pigs but doesn't specify the exact values. Statement 2 gives us a lower bound on the number of pigs but doesn't provide information about the number of cows or the ratio.

Without additional information about the specific number of cows or the ratio between cows and pigs, we cannot determine the exact number of cows.

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

Statement (1): The ratio of P to T is greater than 7:13.
This statement alone is sufficient to answer the question. Since the median of the five numbers is 100 and the range is 60, the middle number (third number) must be 100, and the difference between the first number (P) and the fifth number (T) must be 60. If we assume the ratio of P to T to be 7:13, we can set up the equation:

P + 3d = 100, where d is the common difference between the numbers.
P + 4d = T

By substituting the value of P + 4d = T from statement (1) into the equation P + 3d = 100, we can solve for P and T.

Statement (2): Q is not 80.
This statement alone does not provide enough information to determine the value of T. While it tells us that Q is not 80, it doesn't give us any specific information about T or the other numbers.

By considering statement (1) alone, we can determine the value of T. Therefore, statement (1) alone is sufficient to answer the question.

Hence, the correct answer is (A): Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1): The fisherman who caught the third-most fish caught 11 fish.
This statement alone does not provide enough information to determine whether any one fisherman caught more than 15 fish. We don't know how many fish the first and second most productive fishermen caught, so we cannot make a definitive conclusion.

Statement (2): The fisherman who caught the second-most fish caught 12 fish.
This statement alone also does not provide enough information to determine whether any one fisherman caught more than 15 fish. We still don't know the number of fish caught by the most productive fisherman or any other fishermen besides the second-most productive one.

By combining both statements, we can determine the answer. Since the fisherman who caught the third-most fish caught 11 fish, and the fisherman who caught the second-most fish caught 12 fish, we know that the most productive fisherman must have caught more than 15 fish. Therefore, each statement alone is sufficient to answer the question.

Hence, the correct answer is (D): EACH statement ALONE is sufficient to answer the question asked.

Statement 1: The team won 60% of its first 65 games last season.
This statement tells us the winning percentage of the team in the first 65 games, but it doesn't provide information about the remaining 35 games. Therefore, statement 1 alone is not sufficient to determine if the team won at least half of the games.

Statement 2: The team won 60% of its last 65 games last season.
This statement tells us the winning percentage of the team in the last 65 games, but it doesn't provide information about the first 35 games. Therefore, statement 2 alone is not sufficient to determine if the team won at least half of the games.

When we consider both statements together, we have information about the winning percentage in the first 65 games and the last 65 games. However, we still don't know the winning percentage in the remaining 30 games. Without information about the remaining games, we cannot determine if the team won at least half of the games.

Therefore, the answer is option E: Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Statement 1: The drawer has fewer packets containing blue markers than those containing red markers.
This statement gives us a comparison between the number of packets containing blue markers and those containing red markers. However, it doesn't provide any specific information about the number of packets containing blue markers or their relationship with the packets containing black markers. Therefore, statement 1 alone is not sufficient to determine the number of packets containing a blue marker.

Statement 2: If Josh withdraws packets without looking at their contents, he needs to draw a minimum of 20 packets to ensure that he has exactly 8 markers of any single color out of red, blue, and black.
This statement provides information about the minimum number of packets Josh needs to draw to ensure he has exactly 8 markers of any single color. However, it doesn't directly give us information about the number of packets containing a blue marker. Therefore, statement 2 alone is not sufficient to determine the number of packets containing a blue marker.

When we consider both statements together, we still don't have enough information to determine the exact number of packets containing a blue marker. Statement 1 tells us that there are fewer packets containing blue markers than those containing red markers, but it doesn't provide any information about the number of packets containing black markers. Statement 2 gives us information about the minimum number of packets Josh needs to draw to ensure 8 markers of any single color, but it doesn't specify the distribution of colors within those packets.

Without additional information about the relationship between the packets containing black markers and the packets containing blue markers, we cannot determine the exact number of packets containing a blue marker.

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