Test: Fractions/Ratios/Decimals - GMAT MCQ

(1) The ratio of the number of hardbacks sold to the number of paperbacks sold is 5 to 3.

From this statement, we know the ratio of the number of hardbacks to the number of paperbacks sold. However, we don't have the actual quantities of hardbacks and paperbacks sold, so we cannot calculate the average price based solely on this information. Therefore, statement (1) alone is not sufficient to answer the question.

(2) A total of 160 books were sold during the day.

From this statement, we know the total number of books sold during the day, which is 160. However, we don't have information about the specific quantities of hardbacks and paperbacks sold, so we cannot determine the average price based solely on this information. Therefore, statement (2) alone is not sufficient to answer the question.

By analyzing each statement individually, we can see that neither statement alone is sufficient to determine the average price for all the books sold. However, by combining both statements, we have the ratio of hardbacks to paperbacks sold and the total number of books sold. With this information, we can calculate the average price.

Let's say the number of hardbacks sold is 5x and the number of paperbacks sold is 3x (according to the ratio in statement 1). Therefore, the total number of books sold is 5x + 3x = 8x (according to statement 2). We know that 8x = 160 (according to statement 2), so x = 20. Therefore, the number of hardbacks sold is 5x = 5 * 20 = 100, and the number of paperbacks sold is 3x = 3 * 20 = 60.

The total sales revenue can be calculated as (100 * $20) + (60 * $15) = $2000 + $900 = $2900. The average price for all the books sold can be calculated as $2900 / 160 = $18.125.

Therefore, both statements together are sufficient to determine the average price for all the books sold.

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

(1) The hundreds digit of s is 3.

From this statement, we know the hundreds digit of s is 3. However, we don't have any information about the tens digit from this statement alone. For example, s could be 301, 302, 303, and so on. Therefore, statement (1) alone is not sufficient to answer the question.

(2) The hundreds digit of 10s is 4.

From this statement, we know that the hundreds digit of 10s is 4. This means the tens digit of s is either 3 or 4. However, we don't have enough information to determine the specific tens digit of s. For example, s could be 34, 44, 304, 404, and so on. Therefore, statement (2) alone is not sufficient to answer the question.

By analyzing each statement individually, we can see that neither statement alone is sufficient to determine the tens digit of the positive integer s.

However, if we consider both statements together, we can conclude that the hundreds digit of s is 3 (from statement 1) and the hundreds digit of 10s is 4 (from statement 2). This means the tens digit of s must be 4, as there is no other way to satisfy both statements simultaneously. Therefore, both statements together are sufficient to 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.

(1) The ratio of the total number of members who left at the end of the year to the total number of members at the beginning of the year was 1:6.

From this statement, we know the ratio of members who left at the end of the year to the total number of members at the beginning of the year is 1:6. However, we don't have specific information about the actual number of members who left or the total number of members at the beginning of the year. Therefore, statement (1) alone is not sufficient to answer the question.

(2) At the end of the year, 21 members remained on the Planning Committee.

From this statement, we know the number of members remaining on the Planning Committee at the end of the year is 21. However, we don't have any information about the Finance Committee or the total number of members at the beginning of the year. Therefore, statement (2) alone is not sufficient to answer the question.

When considering each statement individually, neither statement alone is sufficient to determine the number of members in the Finance Committee at the beginning of the year.

However, if we consider both statements together, we can deduce the answer. From statement (1), we know the ratio of members who left at the end of the year to the total number of members at the beginning of the year is 1:6. If we assume that 6x members were in the Finance Committee at the beginning of the year, where x is a positive integer, then the total number of members at the beginning of the year would be 12x (since there were n members in each committee). From statement (2), we know that 21 members remained on the Planning Committee at the end of the year. Therefore, the number of members who left the Planning Committee is 3x (12x - 21).

Since the total number of members who left at the end of the year is given as 5, we can set up the equation:

5 = 3x + 5x
5 = 8x
x = 5/8

Substituting x = 5/8 into the equation for the Finance Committee, we find:

Number of members in the Finance Committee at the beginning of the year = 6x = 6 * (5/8) = 15/4

The number of members must be a positive integer, so we can conclude that the Finance Committee had 4 members at the beginning of the year.

Therefore, both statements together are sufficient to answer the question.

The answer is (D) EACH statement ALONE is sufficient to answer the question asked.

(1) The units digit of y² = 9.

From this statement, we know that the units digit of y² is 9. However, this information alone does not provide us with enough information to determine the units digit of y itself. For example, y could be 3 or 7, both of which satisfy the condition that the units digit of y² is 9. Therefore, statement (1) alone is not sufficient to answer the question.

(2) The units digit of y⁴ = 1.

From this statement, we know that the units digit of y⁴ is 1. However, similar to statement (1), this information alone does not provide us with enough information to determine the units digit of y itself. For example, y could be 1 or 9, both of which satisfy the condition that the units digit of y⁴ is 1. Therefore, statement (2) alone is not sufficient to answer the question.

By analyzing each statement individually, we can see that neither statement alone is sufficient to determine the units digit of y.

When considering both statements together, we have conflicting information. Statement (1) suggests that the units digit of y² is 9, while statement (2) suggests that the units digit of y⁴ is 1. These two conditions cannot be simultaneously satisfied by any single value of y. Therefore, statements (1) and (2) together are not sufficient to answer the question.

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

Statement (1): 1/2 < z < 2/3

From this statement, we can determine that z is greater than 1/2 but less than 2/3. However, this range includes multiple possibilities: 3/5 and 2/3 both satisfy this condition. Therefore, statement (1) alone is not sufficient to determine the value of z.

Statement (2): 5/9 < z < 5/6

Similarly, this statement provides a range for z, but it also includes multiple possibilities: 3/5 and 5/8 both satisfy this condition. Thus, statement (2) alone is not sufficient to determine the value of z.

Since neither statement alone is sufficient to determine the value of z, and when combined they still do not provide enough information to narrow down the possibilities, the correct answer is E: Statements (1) and (2) together are not sufficient to answer the question asked, and additional data is needed.

Statement (1): The ratio of the average weight of all the boys to the average weight of all the girls is 3:2. This statement provides information about the relative weights of boys and girls, but it doesn't provide any information about the total number of boys or girls or their individual weights. Therefore, it is not sufficient to determine the average weight of all the students.

Statement (2): The number of boys and girls in the class are 40 and 60, respectively. This statement provides information about the total number of boys and girls in the class but doesn't provide any information about their weights. Knowing the number of students alone is not enough to calculate the average weight of all the students.

Since neither statement alone provides sufficient information, we cannot determine the average weight of all the students in the class by using either statement alone. 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) states that there are 200 fish in the aquarium. However, this information alone does not provide any specific details about the distribution or ratio of the different types of fish in the aquarium. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2) states that 45 percent of the fish are swordtails, and there are twice as many swordtails as there are angelfish. From this information, we can determine the ratio of swordtails to angelfish, but it doesn't provide any direct information about the number of guppies or their ratio to angelfish. Therefore, statement (2) alone is not sufficient to answer the question.

Considering both statements together, we know from statement (1) that there are 200 fish in the aquarium. From statement (2), we know that 45 percent of the fish are swordtails and there are twice as many swordtails as angelfish.

Since statement (2) provides the information about the ratio of swordtails to angelfish, we can determine the number of angelfish in the aquarium. If the number of angelfish is represented by A, then the number of swordtails is 2A. The total number of angelfish and swordtails is A + 2A = 3A.

We also know from statement (1) that there are 200 fish in total. Therefore, 3A = 200.

Solving this equation, we find A = 66.67, which is not a whole number. This indicates that the ratio of guppies to angelfish cannot be determined with certainty, as it depends on the exact number of fish in the aquarium.

Therefore, when both statements are considered together, they are still not sufficient to answer the question.

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

Statement (1) states that if the amount of liquid A were doubled, the ratio would be 1:2:3. Currently, the ratio of A:B:C is given as 1:4:6. If A were doubled, the new ratio would be 2:4:6, which simplifies to 1:2:3.

From this information, we can determine the volume of liquid A in the mixture. Since the ratio of A to the total mixture volume is 1:(1+4+6) = 1:11, and the new ratio after doubling A is 1:2, we can set up the equation:

2/1 = 11/V, where V represents the total volume of the mixture.

Solving this equation, we find V = 22 liters. Therefore, statement (1) alone is sufficient to answer the question.

Statement (2) states that if 5 liters of liquid A were added, the ratio would be 1:2:3. Since the original ratio of A:B:C is 1:4:6, adding 5 liters of A would change the ratio to (1+5):(4):(6), which simplifies to 6:4:6.

This new ratio does not allow us to determine the exact volume of the mixture since the quantities of B and C remain the same. Therefore, statement (2) alone is not sufficient to answer the question.

Considering both statements together, we have the information from statement (1) that the ratio would be 1:2:3 if the amount of liquid A were doubled, and from statement (2) that the ratio would be 1:2:3 if 5 liters of liquid A were added.

Both statements are consistent with each other, but they still do not provide enough information to determine the total volume of the mixture. Therefore, when both statements are considered together, they are not sufficient to answer the question.

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

(1) In club Y, the number of female members is one less than the number of male members.

From this statement, we know that the number of females is one less than the number of males. However, this information alone does not provide any specific ratios or quantities of males and females. We cannot determine the exact ratio based on this statement alone. For example, if there are 2 males, there would be 1 female, resulting in a ratio of 2:1. If there are 4 males, there would be 3 females, resulting in a ratio of 4:3. Therefore, statement (1) alone is not sufficient to answer the question.

(2) In club Y, the number of male Republican members is equal to the number of female Democratic members.

From this statement, we know that the number of male Republicans is equal to the number of female Democrats. However, we don't have any information about the total number of males or females or the ratio between them. Without knowing the total quantities, we cannot determine the ratio of males to females. Therefore, statement (2) alone is not sufficient to answer the question.

By analyzing each statement individually, we can see that neither statement alone is sufficient to determine the ratio of the number of males to the number of females in club Y.

However, if we consider both statements together, we can conclude that the number of female members is one less than the number of male members (from statement 1) and the number of male Republican members is equal to the number of female Democratic members (from statement 2). From this information, we can determine that the ratio of males to females is 2:1. For example, if there are 4 males, there would be 2 females. If there are 6 males, there would be 3 females. Therefore, statement (2) alone is sufficient to 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) states that the total number of women and children in the room is 12. However, this statement alone does not provide enough information to determine the individual quantities of women, men, and children, or the total number of people in the room. For example, there could be 10 women and 2 children, or 8 women and 4 children, among other possibilities. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2) states that there are fewer than 4 men in the room. This information alone also does not provide enough information to determine the total number of people or the quantities of women, men, and children. We only have information about the upper limit of the number of men, but we don't know the actual quantity. Therefore, statement (2) alone is not sufficient to answer the question.

However, if we consider both statements together, we can deduce the information we need. From statement (1), we know that the total number of women and children is 12. From the given ratio of 5:2:7 for women, men, and children, respectively, we can infer that the number of women is (5/14) * 12 = 5 and the number of children is (7/14) * 12 = 6. Since the total number of people in the room is the sum of women, men, and children, it would be 5 + 2 + 6 = 13. Therefore, both statements together are sufficient to answer the question.

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