Test: Mixture Problems - GMAT MCQ

Statement (1): Adding 5 liters of water increases the percentage of water by 20%.

This statement alone provides information about the effect of adding 5 liters of water, but it does not give us the initial percentage of water or the total volume of the solution. Therefore, it is not sufficient to answer the question.

Statement (2): There are 30 liters of solution before any additions or subtractions from the solution.

This statement tells us the initial volume of the solution, but it does not provide any information about the percentage of water in it. Without knowing the initial percentage, we cannot determine the final percentage of water in the solution. Hence, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we can use the information from statement (1) to determine the final percentage of water. However, we still need to know the initial percentage of water to calculate the final percentage accurately. As neither statement alone is sufficient, but together they provide the necessary information, 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): If 3 liters of additives were added to the mixture, benzene would only account for 30% of the total mixture.

This statement alone provides information about the benzene content relative to the additives when 3 liters of additives are added. However, it does not give us any specific information about the quantities of gasoline, benzene, or additives in the original mixture. Therefore, it is not sufficient to answer the question.

Statement (2): (b + g + 4.5)/3 = 3g/2

This equation provides a relationship between the quantities of gasoline (g) and benzene (b) in the mixture. However, it does not provide any information about the quantity of additives or the total quantity of the mixture. Without knowing the specific quantities of gasoline, benzene, and additives, we cannot determine the number of liters of additives in the tank. Hence, statement (2) alone is not sufficient to answer the question.

When we consider both statements together, we have some information about the benzene content relative to additives (from statement 1) and a relationship between gasoline and benzene quantities (from statement 2). However, we still lack the necessary information to determine the actual quantities or solve for the number of liters of additives. 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): In IIT - entrance exam, 10% of girls in the class failed to qualify.

This statement provides information about the girls in the class who failed to qualify for the IIT entrance exam. However, it doesn't give us any specific numbers or percentages regarding the total number of girls or boys in the class. Therefore, statement (1) alone is not sufficient to determine the number of boys in the class.

Statement (2): There are more than 80 boys in the class.

This statement provides information about the number of boys in the class, stating that there are more than 80 boys. However, it doesn't give us any information about the number of girls or the total number of students in the class. Therefore, statement (2) alone is not sufficient to determine the number of boys in the class.

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

There is at least one girl.
The girls who failed to qualify for the IIT entrance exam make up 10% of the total number of girls.
However, we still don't have specific numbers or percentages to determine the exact number of boys or girls in the class. We only know that there are more than 80 boys, but we don't have any information about the girls.

Therefore, when we consider both statements together, we still cannot determine the exact number of boys in the class. 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): x = y

This statement tells us that the percentage of blue paint (x%) is equal to the percentage of green paint (y%). However, it doesn't provide any specific information about the percentage of red paint (z%) or the actual volume of the mixture. Therefore, statement (1) alone is not sufficient to determine the number of gallons of green paint used.

Statement (2): z = 60

This statement tells us that the percentage of red paint (z%) is equal to 60%. However, it doesn't provide any specific information about the percentage of blue or green paint, or the actual volume of the mixture. Therefore, statement (2) alone is not sufficient to determine the number of gallons of green paint used.

Now let's consider both statements together. We know that x = y (from statement 1) and z = 60 (from statement 2). From this information, we can conclude that the mixture is made up of equal parts blue and green paint, and the remaining percentage (100% - x% - y% = 100% - x% - x% = 100% - 2x%) is red paint, which is 60%.

Since we are given that exactly 1 gallon of blue paint and 3 gallons of red paint were used, we can deduce that the volume of green paint used is also 1 gallon, as it is an equal part of the mixture.

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): The grocer mixed 135 kg of the variety of tea that costs $7 per kg.

This statement tells us the weight of one of the tea varieties used in the mixture. However, it doesn't provide any information about the weights or costs of the other two varieties or the total weight of the mixture. Therefore, statement (1) alone is not sufficient to determine the ratio in which the teas were mixed.

Statement (2): The weight of the mixture was 315 kg.

This statement provides information about the total weight of the mixture but doesn't provide any information about the weights or costs of the individual tea varieties used in the mixture. Therefore, statement (2) alone is not sufficient to determine the ratio in which the teas were mixed.

Now let's consider both statements together. From statement (1), we know that the weight of one of the tea varieties is 135 kg. From statement (2), we know that the total weight of the mixture is 315 kg.

However, even when considering both statements together, we still don't have any information about the costs of the tea varieties or the specific ratios in which they were mixed. Therefore, the information provided by both statements is not sufficient to determine the ratio in which the teas were mixed.

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): If 50 liters of alcohol is added, X will contain 60% alcohol.

This statement tells us the effect of adding 50 liters of alcohol to solution X. It states that after the addition, X will contain 60% alcohol. However, it doesn't provide any specific information about the initial percentage of alcohol in solution X or the total volume of solution X. Therefore, statement (1) alone is not sufficient to determine the percentage of alcohol in solution X.

Statement (2): If the volume of water, equivalent to that of the total solution, is added, X will contain 20% alcohol.

This statement tells us the effect of adding water to solution X. It states that if the volume of water is equivalent to that of the total solution, X will contain 20% alcohol. However, it doesn't provide any specific information about the initial percentage of alcohol in solution X or the total volume of solution X. Therefore, statement (2) alone is not sufficient to determine the percentage of alcohol in solution X.

Now let's consider both statements together. From statement (1), we know that if 50 liters of alcohol is added, X will contain 60% alcohol. From statement (2), we know that if the volume of water, equivalent to that of the total solution, is added, X will contain 20% alcohol.

These statements provide conflicting information. Statement (1) suggests that adding more alcohol increases the percentage of alcohol in solution X, while statement (2) suggests that adding water decreases the percentage of alcohol in solution X. Therefore, the information provided by the two statements is contradictory and cannot be reconciled.

Therefore, when we consider both statements together, we still cannot determine the percentage of alcohol in solution X. Hence, the answer is B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement (1): y = 10

This statement tells us the value of y, which represents the number of tons of a mixture that contained 2 percent gravel G. However, it doesn't provide any specific information about the value of x or the total weight of the resulting mixture (z). Therefore, statement (1) alone is not sufficient to determine the value of x.

Statement (2): z = 16

This statement tells us the value of z, which represents the total number of tons of the resulting mixture. However, it doesn't provide any specific information about the values of x or y. Therefore, statement (2) alone is not sufficient to determine the value of x.

Now let's consider both statements together. From statement (1), we know that y = 10. From statement (2), we know that z = 16. We are also given that the resulting mixture is 5 percent gravel G.

We can set up the equation based on the given information:

(0.1x + 0.02y) / z = 0.05

Substituting the values from statements (1) and (2):

(0.1x + 0.02(10)) / 16 = 0.05
(0.1x + 0.2) / 16 = 0.05
0.1x + 0.2 = 0.8
0.1x = 0.8 - 0.2
0.1x = 0.6
x = 0.6 / 0.1
x = 6

Therefore, by considering both statements together, we can determine the value of x, which is 6 tons.

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

Statement (1): After the addition, water in solution B is 24 L more than spirit.

This statement provides information about the difference in volume between water and spirit in solution B after the addition. However, it doesn't give us any specific information about the initial concentrations of spirit and water in solution B or the final concentration of spirit in solution B. Therefore, statement (1) alone is not sufficient to determine the final concentration of spirit in solution B.

Statement (2): Initial amount of solution A is 140 L.

This statement provides information about the initial amount of solution A, but it doesn't provide any information about the concentrations of spirit and water in solution A or the concentrations in solution B. Therefore, statement (2) alone is not sufficient to determine the final concentration of spirit in solution B.

Now let's consider both statements together. From statement (1), we know that water in solution B is 24 L more than spirit after the addition. From statement (2), we know the initial amount of solution A is 140 L.

Since 28 L of solution is taken from A and added to B, the volume of solution B will increase by 28 L. However, the information about the initial concentrations of spirit and water in solution A is missing, so we cannot determine the exact final concentration of spirit in solution B.

Therefore, when we consider both statements together, we still cannot determine the final concentration of spirit in solution B. Hence, the answer is A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1): y = 16

This statement tells us the value of y, which represents the number of ounces of a saline solution that contained 5 percent saline. However, it doesn't provide any specific information about the value of x or the total volume of the resulting solution (z). Therefore, statement (1) alone is not sufficient to determine the value of x.

Statement (2): z = 30

This statement tells us the value of z, which represents the total number of ounces of the resulting saline solution. However, it doesn't provide any specific information about the values of x or y. Therefore, statement (2) alone is not sufficient to determine the value of x.

Now let's consider both statements together. From statement (1), we know that y = 16. From statement (2), we know that z = 30. We are also given that the resulting solution is 12 percent saline.

We can set up the equation based on the given information:

0.2x + 0.05y = 0.12z

Substituting the values from statements (1) and (2):

0.2x + 0.05(16) = 0.12(30)
0.2x + 0.8 = 3.6
0.2x = 2.8
x = 14

Therefore, by considering both statements together, we can determine the value of x, which is 14 ounces.

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

Statement (1): Water is increased by 14%.

If the water in the container is increased by 14%, we can calculate the new volume of water. However, this statement alone does not provide any information about the alcohol or the total volume of the mixture. Therefore, we cannot determine the new volume of the liquid in the container based solely on this statement.

Statement (2): A mixture whose ratio of water to alcohol is 4:5 is added to that in the container.

This statement provides information about a new mixture that is added to the container, but it doesn't specify the volume of the new mixture or how it relates to the original mixture. Without knowing the volume of the added mixture or how it combines with the existing mixture, we cannot determine the new volume of the liquid in the container based solely on this statement.

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