GMAT Math : Data-Sufficiency Questions

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Example Questions

Example Question #143 : Geometry

Is $\textrm{ polygon } ABCDE$ a regular pentagon?

Statement 1: $m \angle A > m \angle B$

Statement 2: CD > DE

Possible Answers:

BOTH statements TOGETHER are insufficient 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.

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

All of the interior angles of a regular polygon are congruent, as are all of its sides. Statement 1 violates the former condition; statement 2 violates the latter.

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Example Question #7 : Dsq: Calculating An Angle In A Polygon

Thing

Note: Figure NOT drawn to scale.

The diagram above shows a triangle and a rhombus sharing a side. Is that rhombus a square?

Statement 1: The triangle is not equilateral.

Statement 2: $m \angle AVC = 150^{\circ }$

Possible Answers:

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.

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

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

To show that the rhombus is a square, you need to demonstrate that one of its angles is a right angle - that is, $90 ^{\circ }$ . Both statements together are insufficent - if $m \angle AVC = 150^{\circ }$ , you would need to demonstrate that $m \angle AVD = 60^{\circ }$ is true or false, and the fact that the triangle is not equilateral is not enough to prove or to disprove this.

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Example Question #8 : Dsq: Calculating An Angle In A Polygon

Hexagon

Note: Figure NOT drawn to scale.

$m \angle A = m \angle C = m \angle E = 119$

What is $m \angle D$ ?

Statement 1: $m \angle B$ , $m \angle D$ , and $m \angle F$ are the first three terms, in order, of an arithmetic sequence.

Statement 2: $m \angle B + 2 = m \angle F$

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 sufficient to answer the question, but NEITHER 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.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

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

Explanation:

The sum of the measures of the angles of a hexagon is $180 (6-2) = 720^{\circ }$

Therefore,

$m \angle A + m \angle B + m \angle C + m \angle D + m \angle E + m \angle F = 720$

$\ 119 + m \angle B + 119 + m \angle D + 119 + m \angle F = 720$

$m \angle B + m \angle D + m \angle F + 357 = 720$

$m \angle B + m \angle D + m \angle F = 720 - 357 = 363$

Suppose we only know that $m \angle B$ , $m \angle D$ , and $m \angle F$ are the first three terms of an arithmetic sequence, in order. Then for some common difference ,

$m \angle B = m \angle D - d$

$\ m \angle F = m \angle D + d$

$\ \left (m \angle D - d \right )+ m \angle D \left + ( m \angle D +d \right ) = 363$

$\ 3 \cdot m \angle D \left = 363$

$m \angle D \left = 121$

Suppose we only know that $m \angle B + 2 = m \angle F$ . Then

$m \angle B + m \angle D + \left (m \angle B + 2 \right ) = 363$

$2 \cdot m \angle B + m \angle D + 2 = 363$

$2 \cdot m \angle B + m \angle D = 361$

With no further information, we cannot determine $m \angle D$ .

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Example Question #3 : Dsq: Calculating An Angle In A Polygon

Is the degree measure of an exterior angle of a regular polygon an integer?

Statement 1: The number of sides of the polygon is a factor of 30.

Statement 2: The number of sides of the polygon is a factor of 40.

Possible Answers:

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.

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.

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

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The sum of the degree measures of the exterior angles, one per vertex, of any polygon is 360, so each exterior angle of a regular polygon with sides measures $\frac{360}{N}$ . Therefore, the measure of one exterior angle of a regular polygon is an integer if and only if is a factor of 360.

If is a factor of a factor of 360, however, then is a factor of 360. 30 and 40 are both factors of 360: $360 \div 30 = 12$ and $360 \div 40 = 9$ . Therefore, it follows from either statement that the number of sides is a factor of 360, and each exterior angle has a degree measure that is an integer.

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Example Question #10 : Dsq: Calculating An Angle In A Polygon

Is the degree measure of an exterior angle of a regular polygon an integer?

Statement 1: The number of sides of the polygon is divisible by 7.

Statement 2: The number of sides of the polygon is divisible by 10.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

EITHER 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.

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.

Correct answer:

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

Explanation:

The sum of the degree measures of the exterior angles, one per vertex, of any polygon is 360, so each exterior angle of a regular polygon with sides measures $\frac{360}{N}$ .

For $\frac{360}{N}$ to be an integer, every factor of must be a factor of 360. This does not happen if 7 is a factor of , so Statement 1 disproves this. This may or may not happen if 10 is a factor of - $\frac{360}{40} = 9$ , but $\frac{360}{80} = 4.5$ , so Statement 2 does not provide an answer.

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Example Question #11 : Dsq: Calculating An Angle In A Polygon

A nonagon is a nine-sided polygon.

Is Nonagon ABCDEFGHI regular?

Statement 1: $m \angle A = 150 ^{\circ }$

Statement 2: $m \angle G = 150 ^{\circ }$

Possible Answers:

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.

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

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Each angle of a regular nonagon measures $\left [\frac{180 (9-2) }{9 } \right ]^ {\circ } = 140^ {\circ }$

Therefore, each of the two statements proves that the nonagon is not regular.

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Example Question #14 : Polygons

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.

Possible Answers:

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.

Correct answer:

EITHER statement ALONE is sufficient 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.

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Example Question #267 : Data Sufficiency Questions

Pentagon

Note: Figure NOT drawn to scale.

Refer to the above figure. Is Pentagon PENTA regular?

Statement 1: $m \angle ENY = 72 ^{\circ }$

Statement 2: $m \angle TAX = 70 ^{\circ }$

Possible Answers:

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

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 insufficient to answer the question.

Correct answer:

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

Explanation:

If a shape is regular, that means that all of its sides are equal. It also means that all of its interior angles are equal. Finally, if all of the interior angles are equal, then the exterior angles will all be equal to each other as well.

In a regular polygon, we can find the measure of ANY exterior angle by using the formula

$\frac{360}{n}$

where is equal to the number of sides.

Each exterior angle of a regular (five-sided) pentagon measures

$\left (\frac{360 }{5 } \right)^ {\circ } = 72^ {\circ }$

Statement 1 alone neither proves nor disproves that the pentagon is regular. We now know that one exterior angle is $72^{\circ}$ , but we do not know if any of the other exterior angles are also $72^ {\circ }$ .

Statement 2, however, proves that the pentagon is not regular, as it has at least one exterior angle that does not have measure $72^ {\circ }$ .

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Example Question #261 : Data Sufficiency Questions

Thingy_4

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

Statement 1: CD = 8

Statement 2: EB = 15

Possible Answers:

xxx

Either alone

Correct answer:

Either alone

Explanation:

Refer to the figure below, in which $\overline{EB}$ has been constructed, and the top and right sides of the figure have been extended to their intersection to form Rectangle AFDE .

Thingy_4

Assume Statement 1 alone. Since opposite sides of a rectangle have the same length, FD =EA= 12 . By segment addition, FC+CD= FD , and, since CD = 8 , by substitution, FC+8= 12 . Therefore, FC = 4 , and the Pythagorean Theorem can be used to find :

$BF = \sqrt{(BC)^{2} - (FC)^{2}} = \sqrt{5^{2} - 4^{2}} = \sqrt{25-16} = \sqrt{9} = 3$

Since opposite sides of a rectangle have the same length, AF = ED = 12 , and by segment addition, AB + BF = AF . By substitution, AB + 3= 12 , and AB = 9 .

Assume Statement 2 alone. Since $\overline{EB}$ , the hypotenuse of right triangle $\bigtriangleup ABE$ , and $\overline{A E}$ , one of its legs, have lengths 15 and 12, respectively, the length of the other leg $\overline{A B}$ can be found using the Pythagorean Theorem:

$AB = \sqrt{(EB)^{2} - (AE)^{2}} = \sqrt{15^{2} - 12^{2}} = \sqrt{225-144} = \sqrt{81} = 9$

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Example Question #21 : Polygons

Thingy

Note: Figure NOT drawn to scale

Refer to the above figure. Give the length of $\overline{BC}$ .

Statement 1: AB = 7

Statement 2: CD = 12

Possible Answers:

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.

EITHER 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:

We can construct perpendicular line segments from to $\overline{ED}$ and from to $\overline{AE}$ as follows:

Thingy

$\overline{BC}$ is the hypotenuse of a right triangle $\bigtriangleup BGC$ , so if we can determine the lengths of $\overline{BG}$ and $\overline{GC}$ , we can use the Pythagorean Theorem to determine the length of $\overline{BC}$ .

Assume Statement 1 alone. By segment addition, EF+FD = ED = 12 . Since $\overline{AB}$ and $\overline{EF}$ are opposite sides of a rectangle, AB = EF ; similarly, GC = FD . It follows by substitution that AB+GC = 12 . Since AB = 7 , it follows that 7+GC = 12 , and GC = 5 . However, no additional information exists to find .

Assume Statement alone. By similar reasoning, BG + CD = 20 ; since CD = 12 , BG + 12= 20 , and BG = 8 . However, no information exists to find .

The two statements put together, however, yield both necessary values: BG = 8 and GC = 5 . By the Pythagorean Theorem,

$BC = \sqrt{(BG)^{2}+(GC)^{2}} = \sqrt{8^{2}+5^{2}} = \sqrt{64+25} = \sqrt{89}$ .

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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 #143 : Geometry

Example Question #7 : Dsq: Calculating An Angle In A Polygon

Example Question #8 : Dsq: Calculating An Angle In A Polygon

Example Question #3 : Dsq: Calculating An Angle In A Polygon

Example Question #10 : Dsq: Calculating An Angle In A Polygon

Example Question #11 : Dsq: Calculating An Angle In A Polygon

Example Question #14 : Polygons

Example Question #267 : Data Sufficiency Questions

Example Question #261 : Data Sufficiency Questions

Example Question #21 : Polygons

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