Data-Sufficiency Questions

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GMAT Quantitative › Data-Sufficiency Questions

Questions 1 - 10

A line includes points and . Is the slope of the line positive, negative, zero, or undefined?

Statement 1:

Statement 2:

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

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

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

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

Explanation

The slope of the line that includes points and is $m = \frac{b-d}{a -c }$ .

For the question of the sign of the slope to be answered, it must be known whether and are of the same sign or of different signs, or whether one of them is equal to zero.

Statement 1 alone does not answer this question, as it only states that the denominator is greater; it is possible for this to happen whether both are of like sign or unlike sign. Statement 2 only proves that - that is, that the denominator is positive.

If the two statements together are assumed, we know that . Since both the numerator and the denominator are positive, the slope of the line must be positive.

What is the mean of , , , , , and ?

Statement 1:

Statement 2:

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

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

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

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

Explanation

The mean of a data set requires you to know the sum of the elements and the number of elements; you know the latter, but neither statement alone provides any clues to the former.

However, if you know both, you can add both sides of the equations as follows:

$\underline{2a + 4b+6c + 5d + 7e + 3f = 8,000}$

Rewrite as:

and divide by 9:

Now you know the sum, so divide it by 6 to get the mean:

$2,000 \div 6 =333 \frac{1}{3}$ .

1. If the arithmetic mean of five different numbers is 50, how many of the numbers are greater than 50?

(1) None of the five numbers is greater than 100.

(2) Three of the five numbers are 24, 25 and 26, respectively.

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

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

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

EACH statement ALONE is sufficient.

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

Explanation

For statement (1), there are different combinations that satisfy the condition. For example, the five numbers can be $48,49,50,51,\ or\ 52,$ or the five numbers can be . Therefore, we cannot determine how many of the numbers are greater than by knowing the first statement.

For statement (2), even though we know three of them, the two unknown numbers can both be greater than , or one smaller and one greater. Thus statement (2) is not sufficient.

Putting the two statements together, we know that the sum of the two unknown numbers is

$50\cdot 5-\left ( 24+25+26 \right )=175$

Since none of them is greater than 100, both of them have to be greater than 50. Therefore when we combine the two statements, we know that there are two numbers that are greater than 50.

On Monday, 40 people are asked to rate the quality of product A on a seven point scale (1=very poor, 2=poor.....6=very good, 7=excellent).

On Tuesday, a different group of 40 is asked to rate the quality of product B using the same seven point scale.

The results for product A:

7 votes for category 1 (very poor);

8 votes for category 2;

10 votes for 3;

6 vote for 4;

4 votes for 5;

3 votes for 6;

2 votes for 7;

The results for product B:

2 votes for category 1 (very poor);

3 votes for category 2;

4 votes for 3;

6 vote for 4;

10 votes for 5;

8 votes for 6;

7 votes for 7;

It appears that B is the superior product.

Which one of the following statements is true?

The median score for product A is less than the mean score for product A.

The median score of product A is greater than the median score of product B.

The mean score of product A is greater than the mean score of product B.

The median score of product A is greater than the mean score of product A.

The median score of product B is less than the mean score of product B.

Explanation

Median of A = 3

Mean of A = 3.2

Median of B = 5

Mean of B = 4.8

What is the value of in the list above?

(1)

(2) The mode of the numbers in the list is .

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

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

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

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

Each Statement ALONE is sufficient.

Explanation

The mode is the value that appears most often in a set of data. In our list the value that appears most often is n. Therefore n is the mode of the numbers in the list.

Only statement (2) is useful in finding the value of n as it states that the mode of the numbers in the list is 16.

The table in a hall has a length of feet. What is its perimeter?

I) The tabletop is exactly two and a half feet above the floor.

II) The area of the table is four times three more than the width.

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.

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

Both statements are needed to answer the question.

Either statement is sufficient to answer the question.

Explanation

To find perimeter we need length and width. We are given the length.

I) Is irrelevant.

II) We are given a way of relating area and width. Since we know that area is length times width, we can use II to set up an equation where we substitute in the known length along with the given statement to solve for our width

$A=l\cdot w$

$A=4\cdot (w+3)= 14w$

Solve the second one for w and you're good to go!

$\frac{12}{10}=\frac{6}{5}=w$ .

Therefore the perimeter would be:

$P=2l+2w \rightarrow P=2(14)+2\left(\frac{6}{5}\right)$

What is the measure of $\angle 1$ ?

Statement 1: $\angle 1$ is an exterior angle of an equilateral triangle.

Statement 2: $\angle 1$ is an interior angle of a regular hexagon.

EITHER statement ALONE is sufficient 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.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation

An exterior angle of an equilateral triangle measures $360 ^{\circ } \div 3 = 120 ^{\circ }$ . An interior angle of a regular hexagon measures $\left [180 (6-2) \div 6 \right ]^ {\circ} = 120 ^ {\circ}$ . Either statement is sufficient.

Hexagon_44

The hexagon in the above diagram is regular. If $\overline{AB}$ has length 10, which of the following expressions is equal to the length of $\overline{AC}$ ?

$10\sqrt{3}$

$10\sqrt{2}$

$10\sqrt{6}$

$5\sqrt{6}$

Explanation

The answer can be seen more easily by constructing the altitude of $\bigtriangleup ABC$ from , as seen below:

Untitled

Each interior angle of a hexagon measures $120^{\circ }$ ,and the altitude also bisects $\angle ABC$ , the vertex angle of isosceles $\bigtriangleup ABC$ . $\bigtriangleup ABM$ is easily proved to be a 30-60-90 triangle, so by the 30-60-90 Triangle Theorem,

$BM = \frac{1}{2} \cdot AB = \frac{1}{2} \cdot 10 = 5$

$AM = BM \cdot \sqrt{3} = 5\sqrt{3}$

The altitude also bisects $\overline{AC}$ at , so

$AC = 2 \cdot AM = 2 \cdot 5 \sqrt{3} = 10 \sqrt{3}$ .

Find the perimeter of right triangle $\Delta XLC$ .

II) $CX=5\sqrt{2}$

Both statements are needed to answer the question.

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

Either statement is sufficient to answer the question.

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

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

Explanation

If the two shorter sides of a right triangle are equal, that means our other two angles are 45 degrees. This means our triangle follows the ratios for a 45/45/90 triangle, so we can find the remaining sides from the length of the hypotenuse.

I) Tells us we have a 45/45/90 triangle. The ratio of side lengths for a 45/45/90 triangle is $x:x:x\sqrt2$ .

II) Tells us the length of the hypotenuse.

Together, we can find the remaining two sides and then the perimeter.

$CX=5\sqrt2\rightarrow XL=5, CL=5$

$P=5\sqrt2 +5+5$

$P=10+5\sqrt2$

Sector

The circle in the above diagram has center . Give the ratio of the area of the white sector to that of the shaded sector.

Statement 1:

Statement 2: $m \angle ACB = 30 ^{\circ }$

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.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation

We are asking for the ratio of the areas of the sectors, not the actual areas. The answer is the same regardless of the actual area of the circle, so information about linear measurements such as radius, diameter, and circumference is useless. Statement 2 alone is unhelpful.

Statement 1 alone asserts that $m \angle ACB = 30 ^{\circ }$ . $\angle ACB$ is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle $\angle AOB$ that intercepts it - has twice its measure, or $2 \times 30^{\circ }= 60 ^{\circ }$ . From angle addition, this can be subtracted from $180^{\circ }$ to yield the measure of central angle $\angle BOD$ of the shaded sector, which is $120^{\circ }$ . That makes that sector $\frac{120^{\circ }}{360^{\circ }} = \frac{1}{3}$ of the circle. The white sector is $\frac{2}{3}$ of the circle, and the ratio of the areas can be determined to be $\frac{2}{3} : \frac{1}{3}$ , or .