# GMAT Math : Data-Sufficiency Questions

## Example Questions

### Example Question #1 : Dsq: Understanding Diagrams

Four candidates - two boys named Patrick and Quincy, and two girls named Rhonda and Sally - ran for student body president. By the rules, the candidate who wins more than half the ballots cast wins the election outright; if no candidate wins more than half, there must be a runoff between the two top vote-getters. You may assume that no other names were written in.

As can be seen in the figure above, which reflects the share of the vote each candidate won, there will be a runoff. Which two candidates will face each other?

Statement 1: Candidates A and B are both girls.

Statement 2: Candidate B and C are Rhonda and Patrick, respectively.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

Explanation:

Candidates A and B are the top two vote-getters, so we must establish which two candidates are A and B.

Statement 1 does just that by identifying them as Rhonda and Sally. It does not identify which one is which, but it is not necessary to know that.

Statement 2 identifies Rhonda as Candidate B, but only Patrick can be eliminated as Candidate A.

### Example Question #1 : Dsq: Understanding Diagrams

Four candidates - two boys named Mickey and Oswald, and two girls named Nora and Phyllis - ran for student body president. By the rules, the candidate who wins more than half the ballots cast wins the election outright; if no candidate wins more than half, there must be a runoff between the two top vote-getters. You may assume that no other names were written in.

As can be seen in the figure above, which reflects the share of the vote each candidate won, there will be a runoff. Which two candidates will face each other?

Statement 1: Nora is candidate B and Oswald is Candidate C.

Statement 2: Candidates A and C are boys.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

Candidates A and B are the top two vote-getters, so we must establish which two candidates are A and B.

Statement 1 alone establishes that Nora is Candidate B, making her one of the top two vote-getters. Oswald is not one of the top two, since he is Candidate C, but neither Mickey nor Phyllis can be eliminated as Candidate A.

Statement 2 alone established that one boy is Candidate A, one of the top two vote-getters. Since the other boy is Candidate C, then the other one of the top two, Candidate B, is a girl. However, it does not establish the identity of any of them.

Now, assume both statements to be true. By Statement 2, the top two are a boy and a girl. Statement 1 establishes that the girl is Nora. Since Statement 1 also establishes that Oswald is not the boy, the boy is Mickey, and it follows that Mickey and Nora will face each other in the runoff.

### Example Question #1 : Diagrams

Four candidates - Anya, Barry, Carla, and David - ran for student body president. By the rules, the candidate who wins more than half the ballots cast wins the election outright; if no candidate wins more than half, there must be a runoff between the two top vote-getters. You may assume that no other names were written in.

As can be seen in the figure above, which reflects the share of the vote each candidate won, there will be a runoff. Which two candidates will face each other?

Statement 1: Neither Barry nor Carla is Candidate D.

Statement 2: Anya is candidate C.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

Candidates A and B are the top two vote-getters, so we must establish which two candidates are A and B.

Neither statement alone is sufficient to answer the question.

Assume Statement 1 alone. Barry and Carla could be Candidates A and B, respectively, in which case they would be the runoff candidates; also they could be Candidates B and C, respectively, in which case, Barry and one of the other two would be the runoff candidates.

Statement 2 alone only knocks Anya out of the runoff election; it leaves the other three as possible candidates.

Assume both statements to be true. Anya is Candidate C. Candidate D, being neither Barry nor Carla, is David. Therefore, Candidates A and B are Barry and Carla; it is unclear which is which, but it is irrelevant; either way, they are the top two vote-getters, and they will participate in the runoff.

### Example Question #1 : Dsq: Understanding Mixture Problems

Two hydrochloric acid solutions, one of concentration 40% and one of concentration 25%, are mixed together to make a solution of 35% concentration.

1) 200 ml of 40% solution is used

2) 100 ml of 25% solution is used

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is not sufficient.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is not sufficient.

EITHER Statement 1 or Statement 2 ALONE is sufficient to answer the question.

BOTH statements TOGETHER are NOT sufficient to answer the question.

EITHER Statement 1 or Statement 2 ALONE is sufficient to answer the question.

Explanation:

Suppose we only know that 200 ml of the 40% solution is used. Then we can call  the amount of 25% solution used, and the total amount made is  . The solution equation becomes,

So 100 ml of the 25% solution is used.

Similarly, if we only know that 100 ml of the 25% solution is used, then we can call  the amount of 40% solution used, and the total amount made is   . The solution equation becomes,

So 200 ml of the 10% solution is used.

Either way, we get that

and 300 ml of solution is created.

The answer is that either statement is sufficient to answer the question.

### Example Question #1 : Mixture Problems

A delivery truck is carrying two types of packages, green boxes and brown boxes. Green boxes weigh 10 lbs each, while brown boxes are 30 lbs each. At the weigh station, the driver notes that his load weighs exactly 400 lbs. How many green boxes does he have in his truck?

1. He is carrying exactly 20 boxes.

2. The driver never loads more brown boxes than green boxes.

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

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

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

Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

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

Explanation:

Using statement 1, we can find that he has 10 green boxes and 10 brown. We solve mixture problems using 2 equations. If we let  be the number of green boxes, and  be the number of brown boxes. We can set the equations

1)  and

2)

Solving equation 1) for , we get .

Subsituting for  into equation 2) results in

Simplifying we get  or

Therefore, this answer would be 10 green boxes and 10 brown.

The second statement is irrelevant in finding our answer.

### Example Question #1 : Mixture Problems

Mark, the barista at Moose Jaw Coffee, has to mix together two kinds of coffee beans, Chocolate Explosion and Cherry Cherry Delight, to produce a blend that costs $12 per pound. How much of each coffee goes into the mixture? Statement 1: The Chocolate Explosion coffee costs$10 per pound.

Statement 2: The Cherry Cherry Delight coffee costs $16 per pound. Possible Answers: Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question. BOTH statements TOGETHER are insufficient to answer the question. Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question. EITHER statement ALONE is sufficient to answer the question. BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question. Correct answer: BOTH statements TOGETHER are insufficient to answer the question. Explanation: Assume both statements are true. Let be the amount of Chocolate Explosion and be the amount of Cherry Cherry Delight. Then the price of the Chocolate and Cherry coffees will be dollars and dollars, respectively. The total price of the beans is , so we can set up the equation: However, there is no further information we can use to set up a second equation, so there is insufficient information to answer the question. ### Example Question #4 : Dsq: Understanding Mixture Problems Carl, the barista at Moose Jaw Coffee, has to mix together two kinds of coffee beans - Vanilla Heaven, which costs$10 a pound, and Mountain Goodness, which costs $15 a pound - to produce a blend that costs$12 a pound.

How many pounds of Vanilla Heaven coffee beans go into the mixture?

Statement 1: The mixture will include 30 pounds of Mountain Goodness beans.

Statement 2: The finished blend will weigh 75 pounds.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Let  be the number of pounds of Vanilla Heaven coffee beans in the mixture.

If we assume Statement 1 alone, then the total price of the Vanilla Dream coffee beans is $10 per pound times pounds, or . Similarly, the total price of the Mountain Goodness beans is , and the overall price of the beans in the mixture is . Add these together to get the equation . If we assume Statement 2 alone, the price of the Vanilla Dream beans is again . However, the price of the Mountain Goodness beans is and the overall price of the beans is , so we can set up this equation: From either equation we can solve for and get the answer to the question. ### Example Question #5 : Dsq: Understanding Mixture Problems A chemist dilutes liters of a acid solution with water to obtain liters of a acid solution. How many liters of water must be added to the original acid solution? (1) liters (2) liters Possible Answers: Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient Both statements TOGETHER are not sufficient. Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient Each statement ALONE is sufficient. Correct answer: Each statement ALONE is sufficient. Explanation: Using Statement (1), we can set up the following equation: The amount of acid in the original solution (40% multiplied by 15 liters) should equal the amount of acid in the final diluted solution (remember there is no acid in water!!!) Therefore knowing the value of x, we are able to calculate y. We can now find the amount of water used to dilute the solution by subtracting x from y. Using Statement (2), we set up the following equation: We can also calculate the amount of water used since we can calculate x from knowing y. Each Statement ALONE is sufficient to answer the question. ### Example Question #2191 : Gmat Quantitative Reasoning How much should a grocer sell a mixture of almonds and walnuts for? (1) A pound of almonds is twice as expensive as a pound of walnuts. (2) The mixture contains 0.7 pounds of walnuts and 0.3 pounds of almonds. Possible Answers: Each Statement ALONE is sufficient. Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient. Statements (1) and (2) TOGETHER are not sufficient. Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient. Correct answer: Statements (1) and (2) TOGETHER are not sufficient. Explanation: (1) A pound of almonds is twice as expensive as a pound of walnuts. Let x be the price per pound of walnuts and y the price per pound of almonds. We don't know the respective amounts of almonds and walnuts and the prices of any of the nuts aren't clearly stated. Statement(1) Alone is not sufficient. (2) The mixture contains 0.7 pounds of walnuts and 0.3 pounds of almonds. Statement (2) alone is not sufficient, it indicates the quantities of almonds and walnuts in a pound of mixture. However the price per pound of each type of nuts is unknown. Combining both statements, we can write the price of one pound of the mixture as: We do not know x, the price per pound of walnuts, therefore both statements together are not sufficient. ### Example Question #71 : Word Problems What is the price of a pound of almonds? (1) A mixture of half almonds and half walnuts sells for$14 per pound.

(2) A pound of almonds is twice as expensive as the pound of walnuts.

Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient.

Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Each Statement ALONE is sufficient.

Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient.

Statements (1) and (2) TOGETHER are not sufficient.

Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Explanation:

(1) A mixture of half almonds and half walnuts sells for \$14 per pound.

Let x be the price per pound of walnuts and y the price per pound of almonds.

Since we do not know the price per pound of walnuts or the price per pound of almonds, statement(1) is not sufficient.

(2) A pound of almonds is twice as expensive as the pound of walnuts.

Let x be the price per pound of walnuts and y the price per pound of almonds.

Statement (2) alone is not sufficient.

Combining both statements:

The price per pound of almonds is: