Test: Fractions/Ratios/Decimals - GMAT MCQ

Statement (1): If 60% of the members who didn’t bring a guest instead had brought one, then the hall would have been 100% full.
This statement tells us that if 60% of the members who didn't bring a guest had brought one, then the hall would have been completely full. However, it doesn't provide any specific information about the number of members who didn't bring a guest or the total number of seats in the hall. Therefore, statement (1) alone is not sufficient to determine the number of seats.

Statement (2): The number of empty seats was half the number of the AFMS members present.
This statement tells us that the number of empty seats is half the number of AFMS members present. However, it doesn't provide any specific information about the number of members or the total number of seats in the hall. Therefore, statement (2) alone is not sufficient to determine the number of seats.

By considering both statements together, we still cannot determine the number of seats. While statement (1) implies that if 60% of the members brought a guest, the hall would be full, we don't know the exact number or percentage of members who brought guests. Statement (2) tells us that the number of empty seats is half the number of AFMS members present, but we still don't have any specific information about the number of members or guests.

Therefore, when both statements are considered together, we still don't have enough information to determine the number of seats in the AFMS Convention Hall. Additional data is needed.

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

Statement (1): The ratio of the number of children who prefer foxes to the number of children who prefer wolves is 4 to 3.
This statement provides information about the ratio of children who prefer foxes to those who prefer wolves. However, it doesn't provide any specific information about the total number of children surveyed or the number of children who prefer vultures. Without knowing the total number of children surveyed or the ratio of children who prefer vultures, we cannot determine the number of children who prefer foxes. Therefore, statement (1) alone is not sufficient.

Statement (2): The ratio of the number of children who prefer foxes to the total number of children surveyed is 5 to 9.
This statement provides information about the ratio of children who prefer foxes to the total number of children surveyed. However, it doesn't provide any specific information about the number of children who prefer wolves or vultures. Without knowing the number of children who prefer wolves or vultures, we cannot determine the number of children who prefer foxes. Therefore, statement (2) alone is not sufficient.

By considering both statements together, we still cannot determine the number of children who prefer foxes. While statement (1) gives us the ratio of children who prefer foxes to those who prefer wolves, and statement (2) gives us the ratio of children who prefer foxes to the total number of children surveyed, we don't have enough information to calculate the exact numbers or ratios of children who prefer each pet preference.

Therefore, when both statements are considered together, we still don't have enough information to determine the number of children surveyed who prefer foxes. Additional data is needed.

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

To determine the units digit of a positive integer x, we need to consider the remainder when x is divided by 10. This remainder represents the units digit.

Let's evaluate each statement:

Statement (1): The units digit of x/10 = 4.
From this statement, we can deduce that x is a multiple of 10 with a units digit of 4. Therefore, x could be 14, 24, 34, and so on. In each case, the units digit of x is 4. Statement (1) alone is sufficient.

Statement (2): The tens digit of 10x = 5.
From this statement, we can determine that the units digit of x is 5 if the tens digit of 10x is 5. However, the tens digit of 10x depends on the value of x. For example, if x is 15, then the tens digit of 10x is 1. If x is 25, then the tens digit of 10x is 2. The units digit of x cannot be determined from statement (2) alone. Statement (2) alone is not sufficient.

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

To find the product of two single-digit non-zero numbers A and B, we need to determine the values of A and B. Let's evaluate each statement:

Statement (1): When the two-digit number AB is added to BA, the digits at units, tens, and hundreds place in the sum are L, 8, and 1, respectively.
From this statement, we can conclude that A + B = L, A + B + 1 = 8, and A + B + 10 = 1. However, the last equation contradicts the fact that A and B are single-digit numbers. Therefore, statement (1) is not consistent and not sufficient to determine the product of A and B.

Statement (2): When the two-digit number AB is added to BA, the digits at the units place and tens place in the sum differ by 1.
From this statement, we can conclude that A + B = A + B + 1 or B + 1 = 10. This implies that B = 9. However, the value of A is still unknown. Therefore, statement (2) alone is not sufficient to determine the product of A and B.

Since neither statement alone is sufficient to answer the question, the correct answer is B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement (1): Each student received chocolates, biscuits, and toffees in the ratio 1:2:4.
From this statement, we know that the ratio of chocolates to biscuits to toffees is 1:2:4. However, we don't have any specific values for x, y, or z. There are multiple possible combinations that satisfy this ratio, such as x = 1, y = 2, and z = 4 or x = 2, y = 4, and z = 8. Without more information, we cannot determine the exact values of x, y, and z or the number of students in the class. Statement (1) alone is not sufficient.

Statement (2): The class teacher distributed a total of 7 chocolates, 14 biscuits, and 28 toffees, and the number of students present in the class was more than 1.
From this statement, we have specific values for the total number of chocolates, biscuits, and toffees distributed. However, we still don't have enough information to determine the exact values of x, y, and z or the number of students in the class. The given values could be divided among different numbers of students that satisfy the conditions. For example, the values could be divided among 2 students, where each student receives 3 chocolates, 7 biscuits, and 14 toffees. Alternatively, they could be divided among 7 students, where each student receives 1 chocolate, 2 biscuits, and 4 toffees. Statement (2) alone is not sufficient.

Since neither statement alone is sufficient to answer the question, the correct answer is B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement (1): A quarter of the visitors who did not visit the lion's cage were male.
This statement tells us that among the visitors who did not visit the lion's cage, 25% were male. However, it doesn't provide any information about the visitors who did visit the lion's cage or the overall gender distribution of all visitors. Statement (1) alone is not sufficient.

Statement (2): Three quarters of the visitors who visited the lion's cage were female.
This statement tells us that among the visitors who visited the lion's cage, 75% were female. However, it doesn't provide any information about the visitors who did not visit the lion's cage or the overall gender distribution of all visitors. Statement (2) alone is not sufficient.

Combining both statements:
From statement (1), we know that a quarter of the visitors who did not visit the lion's cage were male. From statement (2), we know that three quarters of the visitors who visited the lion's cage were female. However, we still don't have information about the visitors who did visit the lion's cage or the visitors who did not visit the lion's cage. Therefore, even when considering both statements together, we cannot determine the overall gender distribution or calculate the percentage of female visitors. Both statements together are not sufficient.

Therefore, 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): n - p < 0
This statement indicates that n is less than p. However, it doesn't provide direct information about the specific ratio between n and p. We cannot determine if the ratio is 2:1 based on this statement alone.

Statement (2): np = 20
This statement tells us that the product of n and p is 20. Without further information, we cannot determine the individual values of n and p or their ratio. For example, n could be 4 and p could be 5, resulting in a ratio of 4:5, which is not 2:1. Alternatively, n could be 2 and p could be 10, resulting in a ratio of 2:10, which simplifies to 1:5 and is also not 2:1. Statement (2) alone is not sufficient.

Combining both statements:
From statement (1), we know that n - p < 0, which implies that n is less than p. From statement (2), we know that np = 20. Combining these statements, we can deduce that the only possible combination of positive integers n and p that satisfies both conditions is n = 2 and p = 10. In this case, the ratio of n to p is indeed 2:1. Therefore, by considering both statements together, we can determine that the ratio of n to p is 2:1.

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

Statement (1): The sum of the number of cherries and bananas is less than 29.
From the given information, the ratio of apples, bananas, and cherries is 7:5:2. Since the number of cherries is greater than or equal to 5, the possible values for the number of cherries can be 5, 10, 15, and so on. However, we don't have any specific information about the number of bananas or apples, so we cannot determine the number of apples based on this statement alone. Statement (1) alone is not sufficient to answer the question.

Statement (2): The sum of the number of apples and bananas is less than 37.
Similar to the previous statement, this statement provides information about the sum of apples and bananas but doesn't give any specific values for the individual quantities. Therefore, we cannot determine the number of apples based on this statement alone. Statement (2) alone is not sufficient to answer the question.

By considering each statement individually, we find that neither statement alone is sufficient to answer the question.

However, by combining the information from both statements, we can narrow down the possibilities. Since the number of cherries is greater than or equal to 5, and the ratio of apples, bananas, and cherries is 7:5:2, the minimum number of cherries is 5. This means the sum of bananas and cherries is at least 5 + 2 = 7. If the sum of apples and bananas is less than 37, it means the maximum value for the sum of bananas and cherries is 36 - 7 = 29.

Considering the minimum and maximum values, the number of bananas must be between 7 and 29. However, we still don't have enough information to determine the exact number of apples.

Therefore, by considering both statements together, we can conclude that statement (2) alone is sufficient to answer the question. The answer is (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement (1): When x is rounded to the nearest thousandth, the result is 0.455.
This statement tells us the result of rounding x to the nearest thousandth, which is three decimal places. However, since we want to round x to the nearest hundredth, we need to know the digit in the hundredth place. Statement (1) alone does not provide this information, so it is not sufficient to answer the question.

Statement (2): The thousandth digit of x is 5.
This statement provides information about the digit in the thousandth place of x, but it doesn't provide any information about the digits in the hundredth place or beyond. Knowing only the thousandth digit is not enough to determine the result when x is rounded to the nearest hundredth. Statement (2) alone is not sufficient to answer the question.

By considering each statement alone, we find that neither statement is individually sufficient to answer the question.

However, when we consider both statements together, we have the necessary information. Statement (1) tells us the result when x is rounded to the nearest thousandth, which is 0.455. Statement (2) tells us that the thousandth digit of x is 5. Combining this information, we can conclude that the digit in the hundredth place of x must be 4 since rounding to the nearest thousandth yields 0.455.

Therefore, each statement alone is sufficient to answer the question, and the answer is (D) EACH statement ALONE is sufficient to answer the question asked.

Statement (1): The numbers of candies, cookies, and toffees that each student received were in the ratio 3:4:5, respectively.
This statement tells us the ratio of candies, cookies, and toffees that each student received, but it doesn't provide any information about the total number of candies, cookies, toffees, or students. Without knowing the total number of items distributed or the specific quantity received by each student, we cannot determine the number of students in the class. Therefore, statement (1) alone is not sufficient.

Statement (2): The teacher distributed a total of 27 candies, 36 cookies, and 45 toffees.
This statement provides information about the total number of candies, cookies, and toffees distributed by the teacher. However, it doesn't provide any information about the ratio or the number of students. Without the ratio of distribution or the specific quantity received by each student, we cannot determine the number of students in the class. Therefore, statement (2) alone is not sufficient.

By considering both statements together, we still cannot determine the number of students. While statement (1) gives us the ratio of distribution and statement (2) gives us the total quantity distributed, we don't have enough information to calculate the exact quantity received by each student or the total number of students.

Therefore, when both statements are considered together, we still don't have enough information to determine the number of students in the class. Additional data is needed.

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