Test: Mixture Problems - GMAT MCQ

Statement (1): The ingredients A, B, C, and D are in the ratio 10:5:4:2, respectively.
This statement provides the ratio of the ingredients but does not give us any specific quantities. Without knowing the actual quantities, we cannot determine the difference between the amounts of ingredient A and ingredient D. Therefore, statement (1) alone is not sufficient.

Statement (2): The amount (in grams) of ingredient B is 4 more than that of ingredient C.
This statement only provides information about ingredients B and C, but it does not mention anything about ingredients A or D. Without knowing the quantities of ingredients A and D, we cannot determine the difference between them. Hence, statement (2) alone is not sufficient.

Considering both statements together:
While statement (1) gives us the ratio of the ingredients, and statement (2) gives us information about ingredients B and C, we still don't have any specific quantities for ingredients A and D. Therefore, when we combine both statements, we still cannot determine the exact difference in grams between ingredient A and ingredient D. Thus, both statements together are not sufficient.

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) says that the weight of Compound P is always three times the weight of Z.

If the weight of Compound P is always three times the weight of Z, we can set up an equation based on the given ratio:

5g of X : 15g of Y : zg of Z = 25g of X : yg of Y : 3zg of Z

Since the weight of Compound P is always three times the weight of Z, we have:

25g of X + yg of Y + 3zg of Z = 3zg of Z

This equation simplifies to:

25g of X + yg of Y = 2zg of Z

We can see that the equation only involves X, Y, and Z, which are the chemicals mentioned in the problem. Therefore, statement (1) alone is sufficient to answer the question.

Now, let's consider statement (2), which states that the weight of Z is always twice the weight of X.

If the weight of Z is always twice the weight of X, we can set up another equation based on the given ratio:

5g of X : 15g of Y : zg of Z = 25g of X : yg of Y : 2xg of X

Since the weight of Z is always twice the weight of X, we have:

25g of X + yg of Y + 2xg of X = zg of Z

This equation involves X, Y, and Z, which are the chemicals mentioned in the problem. Therefore, statement (2) alone is also sufficient to answer the question.

Both statement (1) and statement (2) individually provide enough information to determine the weight of Z that must be added to maintain the required overall proportion. Therefore, the correct answer is (D) EACH statement ALONE is sufficient to answer the question asked.

Statement (1) says that by weight, the alloy is 3/7 lead and 5/14 copper.

From this information, we can determine the weight of tin in the alloy. Since the alloy is composed of lead, copper, and tin, the remaining weight after accounting for lead and copper would be the weight of tin.

Let's assign variables:
L = weight of lead
C = weight of copper
T = weight of tin

According to statement (1), the alloy is 3/7 lead and 5/14 copper. Therefore, we have the following equations:

L + C + T = total weight of the alloy (56 pounds in this case)
L = (3/7)(total weight of the alloy)
C = (5/14)(total weight of the alloy)

Using these equations, we can solve for the weight of tin, T.

From the given information, we can deduce that statement (1) alone is sufficient to determine the weight of tin in the alloy.

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

To solve this problem, let's analyze each statement:

Statement (1) provides the ratios of petrol and spirit in liquids A and B.

The ratio of petrol to spirit in liquid A is 2:3, and in liquid B, it is 3:4.

However, this statement alone does not provide any information about the composition or ratio of petrol and spirit in liquid C. Therefore, statement (1) alone is not sufficient to answer the question asked.

Now let's consider statement (2). It states that when 20 liters of A, 21 liters of B, and 27 liters of C are mixed, the resulting ratio of petrol to spirit is 29:39.

This statement provides information about the overall mixture of A, B, and C, but it does not specifically give the composition of liquid C. Therefore, statement (2) alone is not sufficient to answer the question asked.

Since neither statement alone is sufficient, let's analyze them together:

Combining both statements, we know the ratios of petrol and spirit in liquids A and B, as well as the overall mixture ratio when A, B, and C are mixed.

However, we still don't have any direct information about the composition or ratio of petrol and spirit in liquid C. Therefore, even when considering both statements together, we cannot determine the percent of petrol in liquid C.

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

1. Let's assume x employees contributed $20 and y contributed $25 then as per the stem 20x + 25y = 250 
2. We can simplify this equation to 4x + 5y = 50 by multiplying both sides with 1/5
3. The question asks us to determine value of x 

Statement-I: The number of employees in the office is 11 (Sufficient)
This states that x + y = 11. Now we have two equations and two unknowns hence we can solve the equations to get the values of x

Statement-II: The number of employees who gave $25 was 1 more than the number of employees who gave $20 (Sufficient)
This states that y = x + 1. Again we have two equations and two unknowns hence we can solve the equations to get the values of x 

Statement 1: To fulfill the ratio of blue : yellow : red = 2 : 3 : 1, we need exactly 20 quarts of the mixture. However, without knowing the availability of yellow paint, we cannot determine the answer to the question. Thus, statement 1 is insufficient.

Statement 2: With the knowledge that exactly 10 quarts of yellow paint are available, we still cannot answer the question definitively without knowing the required volume of the mixed paint. Different volume requirements could result in either enough or insufficient yellow paint. Therefore, statement 2 is insufficient.

Combined statements: When considering both statements together, we know that we need 10 quarts of yellow paint (from statement 1) and have 20 quarts of yellow paint (from statement 2). This guarantees that there is enough yellow paint to satisfy the requirement. Consequently, we can confidently answer the question. Therefore, the combined statements are sufficient.

Statement (1) says that Solution X comprises 3/4 of the combined solution.

This means that the ratio of Solution X to the combined solution is 3:4. However, it does not provide any specific information about the amount of Solution Y or the total volume of the combined solution. Without the quantity of Solution Y, we cannot determine the amount of Solution Y in the new combined solution. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2) states that the combined solution is 16 liters.

This statement provides the total volume of the combined solution. However, it does not provide any information about the composition of the solution or the ratio of Solution X to Solution Y. Without knowing the ratio of Solution X to Solution Y, we cannot determine the amount of Solution Y in the new combined solution. Therefore, statement (2) alone is not sufficient to answer the question.

Combining both statements, we know that Solution X comprises 3/4 of the combined solution, and the total volume of the combined solution is 16 liters. However, we still do not have any specific information about the amount of Solution Y or the ratio of Solution X to Solution Y. Therefore, even when considering both statements together, we cannot determine the amount of Solution Y in the new combined solution.

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

Statement (1) says that last season, The New York Yankees played 160 games.

Knowing the number of games played is helpful, but it does not provide any information about the number of runs scored in each game. Without this information, we cannot determine the average number of runs per game. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2) provides information about the number of runs scored per game in different fractions of their games.

According to statement (2), the Yankees scored four runs per game in exactly 1/5 of their games, five runs per game in exactly 3/4 of their games, and nine runs per game in exactly 1/20th of their games.

This statement gives us the distribution of runs per game in terms of fractions, but it does not provide the actual number of games in each category. Without knowing the number of games falling into each category, we cannot calculate the average number of runs per game. Therefore, statement (2) alone is not sufficient to answer the question.

Combining both statements, we know that the Yankees played 160 games (from statement 1) and we have information about the distribution of runs per game in terms of fractions (from statement 2). However, we still do not have the specific number of games falling into each category. Without this information, we cannot determine the average number of runs per game.

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

Statement (1) says that 3 Gryffindor students play neither chess nor quidditch.

This statement provides information about the number of students who do not play either chess or quidditch. However, it does not directly give us information about the number of students who play both chess and quidditch. Without this specific information, we cannot determine the number of Gryffindor students who play both chess and quidditch. Therefore, statement (1) alone is not sufficient to answer the question.

Statement (2) states that for every 3 Gryffindor students who play neither chess nor quidditch, there are 5 Gryffindor students who play both chess and quidditch.

This statement provides a ratio between the students who play neither chess nor quidditch and those who play both chess and quidditch. However, it does not provide the actual numbers or totals of either group. Without knowing the total number of Gryffindor students or the specific values for either group, we cannot determine the number of students who play both chess and quidditch. Therefore, statement (2) alone is not sufficient to answer the question.

However, when we consider both statements together, we have the information from statement (1) about the number of students who play neither chess nor quidditch and the ratio from statement (2) between the two groups. This allows us to determine the values for both groups and calculate the number of students who play both chess and quidditch.

Therefore, each statement alone is sufficient to answer the question asked.

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

Statement (1): 1.5 hours before they meet, they were 135 miles apart.
From this statement, we can calculate the rate at which Fanny and Alexander are closing the distance between them. In 1.5 hours, Fanny travels 1.5 * 25 = 37.5 miles, and Alexander travels 1.5 * x miles. The total distance they cover in 1.5 hours is 37.5 + 1.5x miles. Since they were 135 miles apart before this time, we can write the equation:

37.5 + 1.5x = 135

Simplifying the equation:

1.5x = 135 - 37.5
1.5x = 97.5
x = 97.5 / 1.5
x = 65

Therefore, statement (1) alone is sufficient to determine the value of x.

Statement (2): y = 360 miles.
This statement provides the initial distance between Fanny and Alexander, but it doesn't give us any information about their rates or the time it takes for them to meet. Without the rate at which they are traveling, we cannot determine the value of x. Therefore, statement (2) alone is not sufficient to determine the value of x.

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