GMAT Math : Data-Sufficiency Questions

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Example Questions

Example Question #122 : Arithmetic

A guitarist grabs a guitar pick out of a dish at random. Find the odds that the pick is green.

I) There are 3 different colors of picks. There are 15 green picks, 45 red picks, and n blue picks.

II) There is a total of 143 picks in the dish.

Possible Answers:

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Both statements are needed to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Either statement is sufficient to answer the question.

Correct answer:

Both statements are needed to answer the question.

Explanation:

A guitarist grabs a guitar pick out of a dish at random. Find the odds that the pick is green.

I) There are 3 different colors of picks. There are 15 green picks, 45 red picks, and n blue picks

II) There is a total of 143 picks in the dish

Recall that probability can be found by the following:

$P=\frac{Desired Outcomes}{Total Outcomes}$

II) Gives us the total number of outcomes

I) Gives us the desired number of outcomes

So our answer can be found by doing the following:

$\frac{15}{143}\approx10.49$

So there is about a 10.49% chance of getting a green pick.

Don't be distracted by the "n" number of blue picks. We still need II) to find the total number of picks, so both are needed.

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Example Question #122 : Arithmetic

Some balls are placed in a large box; the balls include one ball marked "A", two balls marked "B", and so forth up to twenty-six balls marked "Z". A ball is drawn at random.

Given a particular letter of the alphabet, does the probability that that ball will be marked with that letter exceed $\frac{1}{30}$ ?

Statement 1: The letter appears in the word "Barack".

Statement 2: The letter appears in the word "Obama".

Possible Answers:

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

EITHER STATEMENT ALONE provides sufficient information to answer the question.

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

Correct answer:

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

Explanation:

The total number of balls in the box will be

1 + 2 + 3 + ... + 26 .

Since

$1+ 2 + 3 + ...+ N = \frac{N (N + 1)}{2}$ ,

it follows that the number of balls is

$1+ 2 + 3 + ...+ 26= \frac{26 (26 + 1)}{2} = \frac{26 \cdot 27}{2} = 351$ .

The number of balls with a given letter of the alphabet is equal to the number of its position in the alphabet; the probability of a ball with that letter being drawn is that number divided by the total number of balls, 351. Therefore, for this probability to exceed $\frac{1}{30}$ , we must have the relation

$\frac{M}{351} > \frac{1}{30}$

$\frac{M}{351} \cdot 351 > \frac{1}{30} \cdot 351$

$M > \frac{351}{30} = 11 \frac{7}{10}$ .

Therefore, M > 11 .

The 11th letter of the alphabet is "K", so in order to answer this question, it suffices to know whether the letter comes after "K" in the alphabet.

The question cannot be answered from either statement alone; both "Barack" and "Obama" include at least one letter that comes after "K" and at least one that does not. However, if both statements are assumed, since both of the letters shared by the words - "A" and "B" - come before "K", the question can be answered in the negative.

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Example Question #81 : Dsq: Calculating Discrete Probability

Some balls are placed in a large box; the balls include one ball marked "A", two balls marked "B", and so forth up to twenty-six balls marked "Z". A ball is drawn at random.

Given a particular letter of the alphabet, does the probability that that ball will be marked with that letter exceed $\frac{1}{70}$ ?

Statement 1: The letter appears in the word "Mississippi".

Statement 2: The letter does not appear in the word "carbide".

Possible Answers:

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 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.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

The total number of balls in the box will be

1 + 2 + 3 + ... + 26 .

Since

$1+ 2 + 3 + ...+ N = \frac{N (N + 1)}{2}$ ,

it follows that the number of balls is

$1+ 2 + 3 + ...+ 26= \frac{26 (26 + 1)}{2} = \frac{26 \cdot 27}{2} = 351$ .

$\frac{M}{351} > \frac{1}{70}$

$\frac{M}{351} \cdot 351 > \frac{1}{70} \cdot 351$

$M > \frac{351}{70} = 5 \frac{1}{70}$ .

Therefore, M > 5 .

The 5th letter of the alphabet is "E", so in order to answer this question, it suffices to know whether the letter comes after "E" in the alphabet.

Either statement alone is sufficient to answer this question in the affirmative. Statement 1 establishes that the letter must be "I", "M", "P", or "S", all of which come after "E". Statement 2 establishes that the letter cannot be any of "A", "B", "C", "D", or "E", all five of which appear in the word "carbide".

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Example Question #82 : Dsq: Calculating Discrete Probability

Some balls are placed in a large box; the balls include one ball marked "10", two balls marked "9", and so forth up to ten balls marked "1". A ball is drawn at random.

is an integer between 1 and 10 inclusive. True or false: the probability that the ball will have the number marked on it is greater than $\frac{1}{10}$ .

Statement 1: is a prime integer.

Statement 2: $N \le 5$

Possible Answers:

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.

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

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.

Correct answer:

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

Explanation:

The total number of balls in the box will be

1 + 2 + 3 + ... + 10 .

Since

$1+ 2 + 3 + ...+ N = \frac{N (N + 1)}{2}$ ,

it follows that the number of balls is

$1+ 2 + 3 + ...+ 10= \frac{10 (10 + 1)}{2} = \frac{10 \cdot 11}{2} = 55$ .

The frequencies out of 55 of each outcome from 1 to 10, in order, are as follows:

$\left \{ 10, 9, 8, 7, 6, 5, 4, 3, 2, 1\right \}$

Their respective probabilities are their frequencies divided by 55:

$\left \{ \frac{10}{55}, \frac{9}{55}, \frac{8}{55}, \frac{7}{55}, \frac{6}{55}, \frac{5}{55}, \frac{4}{55}, \frac{3}{55}, \frac{2}{55}, \frac{1}{55} \right \}$ .

The probability that the ball will be marked "5" is

$\frac{6}{55} \approx 0.109 > 0.1 = \frac{1}{10}$ ;

therefore, the probability that the ball will be marked with any given integer less than or equal to 5 will be greater than $\frac{1}{10}$ .

The probability that the ball will be marked "6" is

$\frac{5}{55} \approx 0.091 < 0.1 = \frac{1}{10}$ ;

therefore, the probability that the ball will be marked with any given integer greater than or equal to 6 will be less than $\frac{1}{10}$ .

Therefore, it suffices to know whether the number on the ball is less than or equal to 5. Statement 2 states that the number on the ball is less than or equal to 5, so it is sufficient to answer the question in the affirmative. Statement 1 is insufficient, since there are primes less than or equal to 5 - 2, 3, and 5 - and one prime greater than 5, which is 7.

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Example Question #83 : Dsq: Calculating Discrete Probability

Some balls are placed in a large box; the balls include one ball marked "10", two balls marked "9", and so forth up to ten balls marked "1". A ball is drawn at random.

is an integer between 1 and 10 inclusive. True or false: the probability that the ball will have the number marked on it is greater than $\frac{1}{10}$ .

Statement 1: is a perfect square integer.

Statement 2: $N \ge 5$

Possible Answers:

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

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 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.

Correct answer:

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

Explanation:

The total number of balls in the box will be

1 + 2 + 3 + ... + 10 .

Since

$1+ 2 + 3 + ...+ N = \frac{N (N + 1)}{2}$ ,

it follows that the number of balls is

$1+ 2 + 3 + ...+ 10= \frac{10 (10 + 1)}{2} = \frac{10 \cdot 11}{2} = 55$ .

The frequencies out of 55 of each outcome from 1 to 10, in order, is as follows:

$\left \{ 10, 9, 8, 7, 6, 5, 4, 3, 2, 1\right \}$

Their respective probabilities are their frequencies divided by 55:

$\left \{ \frac{10}{55}, \frac{9}{55}, \frac{8}{55}, \frac{7}{55}, \frac{6}{55}, \frac{5}{55}, \frac{4}{55}, \frac{3}{55}, \frac{2}{55}, \frac{1}{55} \right \}$ .

The probability that the ball will be marked "5" is

$\frac{6}{55} \approx 0.109 > 0.1 = \frac{1}{10}$ ;

therefore, the probability that the ball will be marked with any given integer less than or equal to 5 will be greater than $\frac{1}{10}$ .

The probability that the ball will be marked "6" is

$\frac{5}{55} \approx 0.091 < 0.1 = \frac{1}{10}$ ;

therefore, the probability that the ball will be marked with any given integer greater than or equal to 6 will be less than $\frac{1}{10}$ .

Therefore, it suffices to know whether the number on the ball is less than or equal to 5.

Statement 1 alone is insufficient, since there are two perfect square integers from 1 to 5 (1 and 4) and one perfect square integer from 6 to 10 (9). Statement 2 alone is insufficient, since it is not clear whether the number on the ball is 5 or a number greater than 5. However, from the two statements together, it can be inferred that N = 9 , and that the probability of drawing a ball with this number is $\frac{2}{55} < \frac{1}{10}$ .

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Example Question #3221 : Gmat Quantitative Reasoning

A bag contains x red marbles, y blue marbles, and z green marbles. What is the probability of drawing a green marble?

(1) There's a probability of drawing a red marble.

(2) There's a probability of drawing a blue marble.

Possible Answers:

EACH statement ALONE is 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.

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

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

Correct answer:

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

Explanation:

Probability is a part to whole comparison.

With statement 1, we don't know nor can we determine the probability of drawing a green marble because we know nothing about the probability of drawing a blue marble. Therefore, statement 1 is not sufficient.

With statement 2, we don't know nor can we determine anything regarding the probability of drawing a red marble. Therefore, statement 2 is not sufficient.

However, taken together, we can determine that the probability of drawing a green marble is . Therefore, BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

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Example Question #81 : Dsq: Calculating Discrete Probability

A coin is tossed n times. What is the probability of getting at least one tail?

(1) The probability of never flipping a head is $\frac{1}{8}$ .

(2) The probability of flipping at least one head is $\frac{7}{8}$ .

Possible Answers:

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

EACH statement ALONE is sufficient.

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

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

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

Correct answer:

EACH statement ALONE is sufficient.

Explanation:

Probability is the number of desired outcomes divided by the number of possible outcomes.

With statement 1, we are told the probability of never flipping a head is $\frac{1}{8}$ , which is the same as always flipping a tail. Since the probability of flipping a tail on a single flip is $\frac{1}{2}$ , we can determine the probability of flipping n tails is $\frac{1}{2}^{n}=\frac{1}{8}$ , which solves as $\frac{1}{2}^{3}=\frac{1}{8}$ . With n=3, we can solve for the probability of getting at least one tail is 1-probability of never getting a tail or 1-HHH:

$1-\frac{1}{8}=\frac{7}{8}$ .

Therefore, statement 1 alone is sufficient.

With statement 2, we are told the probability of flipping at least one head is $\frac{7}{8}$ . Since there are only 2 possible outcomes with each flip (heads or tails), the probability of flipping at least one head is the same as the probability of flipping at least one tail. Therefore, statement 2 alone is sufficient.

Therefore, the correct answer is EACH statement ALONE is sufficient.

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Example Question #1 : Counting Methods

Oteri can choose to wear jeans, sweat pants, or shorts, a button down or a tee shirt, and a hat or no hat. How many different outfit combinations does he have in his wardrobe?

Possible Answers:

120

Correct answer:

Explanation:

He has 3 pants, 2 shirts, 2 hat choices (hat or no hat).

$3\times 2\times 2= 12$

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Example Question #2 : Counting Methods

Olivia picks balls, each one after the other, from a basket containing balls.

How many possible combinations of balls can she end up with?

(1) x + y = 13

(2) 2x + 2y = 26

Possible Answers:

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

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient

Correct answer:

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

Explanation:

Each statement on its own gives us one equation with 2 unknowns, which is unsolvable. Therefore each statement alone is not sufficient.

The two equations are equivalent: x + y = 13 and 2.(x+y) = 2 x 13 = 26

Therefore we cannot sole for x and y either with both equations.

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Example Question #3 : Counting Methods

A Lunch Deal Meal at the Chinese restaurant where Jack likes to eat comprises one appetizer, one entree, and one beverage. How many possible Lunch Meal Deals can Jack choose from?

Statement 1: Jack can choose from seven appetizers and ten entrees.

Statement 2: Jack can choose from as many beverages as appetizers.

Possible Answers:

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.

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.

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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

By the multiplication principle, the number of meals Jack can make is the product of the number of appetizers, the number of entrees, and the number of beverages. Statement 1 tells only the first two; Statement 2 does not give any of these quantities. From the two statements together, we know that there are seven appetizers, ten entrees, and seven beverages, so there are $7 \times 10 \times 7 = 490$ possible Lunch Deal Meals.

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GMAT Math : Data-Sufficiency Questions

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #122 : Arithmetic

Example Question #122 : Arithmetic

Example Question #81 : Dsq: Calculating Discrete Probability

Example Question #82 : Dsq: Calculating Discrete Probability

Example Question #83 : Dsq: Calculating Discrete Probability

Example Question #3221 : Gmat Quantitative Reasoning

Example Question #81 : Dsq: Calculating Discrete Probability

Example Question #1 : Counting Methods

Example Question #2 : Counting Methods

Example Question #3 : Counting Methods

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