Test: Overlapping Sets - GMAT MCQ

Step 1: Analyse Question Stem
Total drinks sold at Darlene’s Beverage Shop = 58

• Total Carbonated drinks sold =21 
• So, total non – carbonated drinks sold  =58 – 21 = 37 

We need to find out the number of carbonated drinks sold which were caffeinated too.

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE
Statement 1: There are 30 caffeinated drinks sold in Darlene’s Beverage Shop.
According to this statement:
(The number of carbonated drinks sold which were caffeinated) + (the number of non-carbonated drinks sold which were caffeinated)
However, we don’t know the number of non – carbonated drinks sold which were caffeinated.
= 30
Hence, we cannot find out the number of carbonated drinks sold which were caffeinated.
Hence, statement 1 is NOT sufficient and we can eliminate answer Options A and D.

Statement 2: Twenty-two of the non-carbonated drinks sold in Darlene’s Beverage Shop are not caffeinated.

According to this statement:

• Non -carbonated drinks sold which were non -caffeinated = 22
• So, (Non -carbonated drinks sold which were caffeinated) = (total non – carbonated drinks sold) - (Non -carbonated drinks sold which were non -caffeinated) = 37 – 22 = 15 

However, with the above information we cannot find the number of carbonated drinks sold which were caffeinated.
Hence, statement 2 is also NOT sufficient and we can eliminate answer Option B.

Step 3: Analyse Statements by combining.
From statement 1:

• (The number of carbonated drinks sold which were caffeinated) + (the number of non-carbonated drinks sold which were caffeinated) = 30

From statement 2:

• Non -carbonated drinks sold which were caffeinated = 15

On Combining both statements:

• We get, the number of carbonated drinks which were caffeinated = 30 – 15 = 15

Thus, the correct answer is Option C.

Statement (1): 45% of the children have an average score of 80 in their final exams.
From this statement alone, we know that 45% of the children have an average score of 80. However, we don't have any information about the gender distribution within this group or the total number of girls in the group. Statement (1) alone is not sufficient.

Statement (2): 50% of the children in the group are girls.
From this statement alone, we know that 50% of the children in the group are girls. However, we don't have any information about their average scores or the total number of children in the group. Statement (2) alone is not sufficient.

Combining both statements, we have the following information:

• 45% of the children have an average score of 80 in their final exams (Statement 1).
• 50% of the children in the group are girls (Statement 2).

Although we have information about the percentage of children with an average score of 80 (Statement 1) and the percentage of girls in the group (Statement 2), we still don't know the exact numbers or how these two groups overlap.

For example, if there are 200 children in total, Statement 2 tells us that 100 of them are girls. However, we don't know how many of these girls have an average score of 80. Additionally, we don't know the number of boys in the group or the distribution of average scores among them.

Therefore, Statements (1) and (2) TOGETHER are not sufficient to determine the number of girls with an average score of 80 in the group. Additional data is needed to answer the question accurately.

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

Statement (1): 28 of the 40 securities are stocks.
From this statement alone, we know that out of the 40 securities in the portfolio, 28 are stocks. However, we don't have any information about the classification of these stocks as domestic or international. Statement (1) alone is not sufficient.

Statement (2): 35 of the 40 securities are domestic.
From this statement alone, we know that out of the 40 securities in the portfolio, 35 are domestic. However, we don't have any information about the specific classification of these securities as stocks or other types of investments. Statement (2) alone is not sufficient.

Combining both statements, we have the following information:

• 28 of the 40 securities are stocks (Statement 1).
• 35 of the 40 securities are domestic (Statement 2).

Although we have information about the number of stocks and the number of domestic securities, we still don't know the overlap between these two categories. It is possible that all 28 stocks are domestic, or some of them could be international. Similarly, out of the 35 domestic securities, some could be stocks or other types of investments.

Without knowing the specific overlap between stocks and domestic securities, we cannot determine the number of domestic stocks in the portfolio.

Therefore, Statements (1) and (2) TOGETHER are not sufficient to answer the question. Additional data is needed, such as the overlap between stocks and domestic securities or more specific information about the composition of the portfolio.

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

Statement (1): 10% of the alumni students who came were females.
This statement provides information about the gender distribution among the alumni students who attended the meet. However, it doesn't provide any information about the gender distribution among the college studying students. Without this information, we can't determine the overall percentage of males who participated in the meet.

Statement (2): 20% of college studying students who participated were males.
This statement provides information about the gender distribution among the college studying students who participated in the meet. However, it doesn't provide any information about the gender distribution among the alumni students. Without this information, we can't determine the overall percentage of males who participated in the meet.

Since neither statement alone is sufficient to answer the question, we need to evaluate both statements together:

By combining the information from both statements, we know that 10% of the alumni students who attended were females, and 20% of the college studying students who participated were males. However, we still don't have the specific proportions of alumni and college studying students in the total student population. Without this information, we can't determine the overall percentage of males who participated in the meet.

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): 35 of the students are members of both the chess club and the forensic society.
This statement provides information about the overlap between the chess club and the forensic society, but it doesn't give any information about the drama club. Without information about the drama club, we can't determine the number of students who are members of both the chess club and the drama club based on this statement alone.

Statement (2): 45 of the students are members of both the drama club and the forensic society.
This statement provides information about the overlap between the drama club and the forensic society, but it doesn't give any information about the chess club. Without information about the chess club, we can't determine the number of students who are members of both the chess club and the drama club based on this statement alone.

Since neither statement alone is sufficient to answer the question, we need to evaluate both statements together:

Even when we combine the information from both statements, we still don't have any direct information about the overlap between the chess club and the drama club. We only have information about the overlaps between each club and the forensic society. Therefore, we can't determine the number of students who are members of both the chess club and the drama club based on these two statements alone.

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

Statement (1): There are 21 men on the construction site.
This statement alone tells us the number of men, but it doesn't provide any information about the women or the number of people wearing or not wearing helmets. Therefore, we can't determine the total number of people on the site based on this statement alone.

Statement (2): There are 24 people that do not wear helmets on the construction site.
This statement provides information about the number of people not wearing helmets, but it doesn't give any details about the breakdown between men and women or the total number of people. Without additional information, we can't determine the total number of people on the site based on this statement alone.

Since neither statement alone is sufficient to answer the question, we need to evaluate both statements together:

If we combine the information from both statements, we know that there are 21 men and 24 people not wearing helmets. However, we still don't have any information about the number of women or the number of people wearing helmets. Without these details, we can't determine the total number of people on the construction site.

Therefore, the correct answer is (E): Statements (1) and (2) together are not sufficient to answer the question, and additional data are needed.

Statement (1): 25 travelers enjoyed traveling by plane.
From this statement alone, we know that 25 travelers enjoyed traveling by plane. However, we don't have any information about their preferences for traveling by train or the overlap between the two groups. Statement (1) alone is not sufficient.

Statement (2): All of the travelers who enjoyed traveling by plane also enjoyed traveling by train.
From this statement alone, we know that all the travelers who enjoyed traveling by plane also enjoyed traveling by train. This means that the travelers who enjoyed traveling by both plane and train are included in the group of travelers who enjoyed traveling by plane. However, we don't have any information about the total number of travelers or the number of travelers who enjoyed traveling by train only. Statement (2) alone is not sufficient.

Combining both statements, we have the following information:

• 25 travelers enjoyed traveling by plane (Statement 1).
• All of the travelers who enjoyed traveling by plane also enjoyed traveling by train (Statement 2).

From statement (2), we know that the travelers who enjoyed traveling by both plane and train are included in the group of travelers who enjoyed traveling by plane. Since statement (1) tells us that 25 travelers enjoyed traveling by plane, and all of them also enjoyed traveling by train (according to statement 2), we can conclude that the number of travelers who enjoyed traveling by both plane and train is 25.

Therefore, both statements together are sufficient to determine that 25 travelers enjoyed traveling by both plane and train. 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.

Statement (1): There are 40 angelfish in that aquarium.
From this statement alone, we know that there are 40 angelfish in the aquarium, but we don't have any information about the number of goldfish or the male-to-female ratio among the angelfish or goldfish. Statement (1) alone is not sufficient.

Statement (2): There are 25 female fish in that aquarium.
From this statement alone, we know that there are 25 female fish in the aquarium, but we don't have any information about the total number of fish, the species distribution, or the male-to-female ratio among the goldfish or angelfish. Statement (2) alone is not sufficient.

Combining both statements, we have the following information:

• There are 40 angelfish in the aquarium (Statement 1).
• There are 25 female fish in the aquarium (Statement 2).

Although we know the number of angelfish and female fish, we still don't have information about the total number of fish, the number of goldfish, or the male-to-female ratio among the goldfish and angelfish. Without this additional data, we cannot determine the number of male goldfish in the aquarium.

For example, if there are 50 total fish in the aquarium (40 angelfish and 10 goldfish), and all the male fish are goldfish, then the number of male goldfish would be 10. However, if there are 100 total fish (40 angelfish and 60 goldfish), and all the male fish are angelfish, then the number of male goldfish would be 0.

Therefore, Statements (1) and (2) TOGETHER are not sufficient to determine the number of male goldfish in the aquarium. Additional data, such as the total number of fish or the male-to-female ratio among the goldfish and angelfish, is needed to answer the question accurately.

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

(1) Of the biology students, 30 percent are also chemistry students.
So, the overlap between the two numbers is 30% of chemistry students.
But the two numbers are not given.
Not Sufficient.

(2) Of the chemistry students, 40 percent are also biology students.
So, the overlap between the two numbers is 40% of biology students.
But the two numbers are not given.
Not Sufficient.

Combining:
The overlap is 30% of chemistry students and 40% of biology students and since the overlap is the same for both means it constitutes bigger chunk from biology than chemistry.
If mathematical representation makes things easier than here it is:
0.3c = 0.4b
c/b = 4/3
So, chemistry students number must be higher than biology.
C is sufficient.

Statement (1): Exactly 120 of the members are string players.
From this statement alone, we know that there are exactly 120 string players in the community orchestra. However, we don't have any information about the total number of members or the breakdown of male and female string players. Statement (1) alone is not sufficient.

Statement (2): Exactly 35 percent of the players who are not string players are male.
From this statement alone, we know the gender distribution of non-string players, but we don't have any information about the total number of members or the percentage of female string players. Statement (2) alone is not sufficient.

Combining both statements, we have the following information:

• Exactly 120 of the members are string players (Statement 1).
• Exactly 35 percent of the players who are not string players are male (Statement 2).

From Statement 1, we know that there are 120 string players, but we don't have information about the total number of members to calculate the percentage of female string players. However, we can determine the total number of non-string players by subtracting 120 from the total number of members.

From Statement 2, we know the percentage of male non-string players, but we still don't have information about the total number of members or the percentage of female string players.

Therefore, Statement 1 alone is sufficient to answer the question because we can calculate the number of female string players using the given percentage from Statement 2. The percentage of female string players can be determined by subtracting the percentage of male non-string players (35%) from the total percentage of female players (100% - 35%). Once we have the percentage, we can multiply it by the total number of string players (120) to find the number of female string players.

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