Test: Fractions/Ratios/Decimals - GMAT MCQ

Statement (1): c – a = 12

This equation indicates that the difference between c and a is equal to 12. However, it doesn't provide direct information about the values of a or c. Therefore, Statement (1) alone is not sufficient to answer the question.

Statement (2): b – a = 6

This equation indicates that the difference between b and a is equal to 6. While it gives information about the relationship between a and b, it doesn't provide direct information about the values of a or c. Therefore, Statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we have:

Statement (1): c – a = 12
Statement (2): b – a = 6

From Statement (2), we can rewrite it as b = a + 6. Substituting this expression into the ratio, we have:

a : (a + 6) : c = 1 : 3 : 5

We can rewrite this as:

a/(a + 6) = 1/3 and c/(a + 6) = 5/3

To solve these equations, we can set them equal to each other:

a/(a + 6) = c/(a + 6)

Cross-multiplying, we have:

a(a + 6) = c(a + 6)

Expanding, we get:

a^2 + 6a = ac + 6c

Rearranging terms, we have:

a^2 + 6a - ac - 6c = 0

Factoring out a common factor of a, we get:

a(a + 6 - c) - 6c = 0

Since a, b, and c are integers and the ratio is given as 1 : 3 : 5, we can deduce that a = 1, b = 3, and c = 5.

Therefore, a + c = 1 + 5 = 6.

Both statements together are sufficient to determine the value of a + c. Thus, the answer is D: EACH statement ALONE is sufficient to answer the question asked.

Statement (1): y = 1/(5^z)

This statement provides the relationship between y and z. However, it doesn't directly provide any information about the value of x or whether x > 2. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): z has exactly three unique factors and is a positive integer less than 9.

This statement provides information about the factors of z and its value range. However, it doesn't provide any direct information about y, x, or whether x > 2. Therefore, statement (2) alone is not sufficient to answer the question.

Considering both statements together, we have information about the relationship between y and z from statement (1) and information about the factors and value range of z from statement (2). However, we still don't have direct information about x or whether x > 2. Without additional data, we cannot determine the value of x or whether it is greater than 2.

Therefore, both statements together are not sufficient to answer the question.

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

Statement (1): Jack received 2/9 of what Pollock and Gibbs together received.

From this statement, we know the ratio of the amount Jack received to the amount Pollock and Gibbs received together. However, we don't have any specific information about the individual amounts received by any of them. Therefore, statement (1) alone is not sufficient to determine who received the minimum amount.

Statement (2): Pollock received 3/11 of what Jack and Gibbs together received.

This statement provides the ratio of the amount Pollock received to the amount Jack and Gibbs received together. Similar to statement (1), we don't have specific information about the individual amounts received by any of them. Therefore, statement (2) alone is not sufficient to determine who received the minimum amount.

Considering both statements together, we have information about the ratios of amounts received by Jack, Pollock, and Gibbs relative to each other. However, we still don't have enough information to determine the specific amounts received by each of them. The minimum amount can vary depending on the actual values.

Therefore, both statements together are not sufficient to answer the question.

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

Statement (1): 1/3 of the people polled enjoyed cupcakes, 1/2 enjoyed apple pie, and 3/4 enjoyed ice cream sundaes.

Based on this statement, we know the individual fractions of people who enjoyed each dessert. However, we cannot determine the fraction of people who enjoyed exactly two desserts solely based on this information. For example, it is possible that some people enjoyed all three desserts, which would contribute to the fraction of people who enjoyed exactly two desserts. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): 1/4 of the people polled enjoyed all three desserts.

This statement provides information about the fraction of people who enjoyed all three desserts. However, it does not provide any information about the fractions of people who enjoyed only one or two desserts. Without this additional information, we cannot determine the fraction of people who enjoyed exactly two desserts. Therefore, statement (2) alone is not sufficient to answer the question.

Considering both statements together, we have information about the fractions of people who enjoyed each dessert individually (statement 1) and the fraction of people who enjoyed all three desserts (statement 2). However, we still do not have enough information to determine the fraction of people who enjoyed exactly two desserts. The fraction of people who enjoyed exactly two desserts depends on the overlap between the different dessert preferences, which is not specified in the given statements.

Therefore, both statements together are not sufficient to answer the question.

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

Statement (1): Aditi has half as many stamps as John has, and Karim has three times as many stamps as John has.

If we denote the number of stamps John, Aditi, and Karim have as J, A, and K, respectively, according to statement (1), we have A = (1/2)J and K = 3J. To find the ratio of the number of stamps Aditi has to the number of stamps John has, we can substitute A = (1/2)J into the ratio expression:

A/J = (1/2)J/J = 1/2

Therefore, statement (1) alone is sufficient to determine the ratio of the number of stamps Aditi has to the number of stamps John has.

Statement (2): Aditi has 1/6 the size collection of what Karim has.

According to statement (2), we have A = (1/6)K. However, this statement alone does not provide any information about the number of stamps John has, so we cannot determine the ratio of the number of stamps Aditi has to the number of stamps John has based on this statement alone.

Considering both statements together, we have the information A = (1/2)J and A = (1/6)K. Since we also know K = 3J from statement (1), we can substitute these values into the ratio expression:

A/J = ((1/2)J)/J = 1/2

Therefore, even when considering both statements together, we obtain the same ratio as in statement (1) alone.

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

Statement (1): 1/r is a terminating decimal

If 1/r is a terminating decimal, it means that r must be a power of 10 or have only factors of 2 and 5 in its prime factorization. However, knowing this information about r does not provide any direct information about the terminating nature of r/s. So, statement (1) alone is not sufficient to answer the question.

Statement (2): 1/s is a terminating decimal

Similarly, if 1/s is a terminating decimal, it means that s must be a power of 10 or have only factors of 2 and 5 in its prime factorization. But, like in statement (1), this information about s does not directly tell us whether r/s is a terminating decimal or not. Therefore, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we know that both r and s have the properties of being powers of 10 or having only factors of 2 and 5 in their prime factorizations. However, this still does not give us enough information to determine whether r/s is a terminating decimal or not. It is possible for r/s to be a terminating decimal, but it is also possible for it to be a repeating decimal if there are common factors between r and s. Therefore, even when considering both statements together, we cannot answer the question.

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

Statement (1): R = 12

If we know the value of R, we can substitute it into the expression (R + K)/(S + K) and simplify it. However, since we don't have any information about the value of K or S, we cannot determine the value of the entire expression. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): K = 7

If we know the value of K, we can substitute it into the expression (R + K)/(S + K) and simplify it. However, since we don't have any information about the values of R or S, we cannot determine the value of the entire expression. Therefore, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we have additional information. We know the value of R and the value of K. However, we still don't have any information about the value of S. Without the value of S, we cannot determine the value of the entire expression (R + K)/(S + K). Therefore, even when considering both statements together, we cannot answer the question.

Therefore, both statements together are sufficient to determine some parts of the expression, but they are not sufficient to answer the question asked. The answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

Statement (1): The average (arithmetic mean) of b and d is 9.

If the average of b and d is 9, we can write the equation: (b + d) / 2 = 9.

This equation alone does not provide information about the individual values of b and d or their range. It only tells us the average of b and d. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2): The sum of a and d is 15.

If the sum of a and d is 15, we can write the equation: a + d = 15.

This equation alone does not provide information about the individual values of a and d or their range. It only tells us their sum. Therefore, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we have additional information. We know that the ratio of a to b to c to d is 1 to 2 to 3 to 4, and we know the sum of a and d is 15.

From the ratio, we can write the equations: b = 2a, c = 3a, and d = 4a.

Substituting these expressions into the equation a + d = 15, we get:
a + 4a = 15
5a = 15
a = 3

Using this value of a, we can determine the values of b, c, and d:
b = 2a = 2(3) = 6
c = 3a = 3(3) = 9
d = 4a = 4(3) = 12

The range of the values a, b, c, and d is 3 to 12.

Therefore, each statement alone is sufficient to answer the question asked. The answer is D: EACH statement ALONE is sufficient to answer the question asked.

Statement (1): r < 1/3

This statement tells us that the decimal representation of r is less than 1/3. Since r is represented by the decimal 0.t5, where t is the unknown digit, we can conclude that t must be less than 3. However, we still don't know the exact value of t. Statement (1) alone is not sufficient to answer the question.

Statement (2): r < 1/10

This statement tells us that the decimal representation of r is less than 1/10. In this case, we can conclude that t must be less than 1. Since t represents a digit, the only possible value for t in this case is 0. Therefore, we can determine that t = 0. Statement (2) alone is sufficient to answer the question.

When we consider both statements together, we have conflicting information. Statement (1) implies that t < 3, while Statement (2) implies that t = 0. These statements cannot be true simultaneously. Therefore, Statements (1) and (2) together are not sufficient to answer the question.

Based on the above analysis, the answer is B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement (1): Company X employed 20 more women in 1996 than in 1995.

This statement provides information about the change in the number of women employed by Company X between 1995 and 1996. However, it doesn't give any information about the number of men employed or the total number of employees. Therefore, Statement (1) alone is not sufficient to answer the question.

Statement (2): Company X employed 20 more men in 1996 than in 1995.

Similar to Statement (1), this statement only provides information about the change in the number of men employed by Company X between 1995 and 1996. It doesn't provide any information about the number of women employed or the total number of employees. Thus, Statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we still lack crucial information to determine the ratio of men to women employed in 1996. While we know that there was an increase of 20 men and an increase of 20 women compared to 1995, we don't know the initial numbers for men and women in 1995 or the total number of employees in either year. Therefore, Statements (1) and (2) together are not sufficient to answer the question.

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