Data-Sufficiency Questions

Help Questions

GMAT Quantitative › Data-Sufficiency Questions

Questions 1 - 10

Tetra_1

Note: Figure NOT drawn to scale.

Refer to the above figure, which shows a tetrahedron, or triangular pyramid. What is the volume of the tetrahedron?

Statement 1: $\bigtriangleup ABD$ is an isosceles triangle with area 64.

Statement 2: $\bigtriangleup ABC$ is an equilateral triangle with perimeter 48.

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.

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 insufficient to answer the question.

Explanation

Each statement gives enough information about one triangle to determine its area, its angles, and its sidelengths, but no information about the other three triangles is given except for one side.

Assume both statements are known. $\bigtriangleup ABD$ is an isosceles triangle with area 64. Since , we can find this common sidelength using the area formula for a triangle, with these lengths as height and base:

$\frac{1}{2} \cdot AD \cdot BD = 64$

$AD \cdot AD = 128$

$(AD)^{2} = 128$

$AD = \sqrt{128} =8 \sqrt{2}$ .

This is the length of both $\overline{AD}$ and $\overline{BD}$ .

By the 45-45-90 Theorem, $\overline{AB}$ has length $\sqrt{2}$ times this, or $8 \sqrt{2} \cdot \sqrt{2} = 8 \cdot 2 = 16$ .

Since $\bigtriangleup ABC$ is an equilateral triangle, . Since $\bigtriangleup ADC$ is a right triangle, , and $AD =8 \sqrt{2}$ , the triangle is also isosceles, and $CD= 8\sqrt{2}$ ; by a similar argument, $BD= 8\sqrt{2}$ .

The volume of the pyramid can be calculated. Its base, which is congruent to $\bigtriangleup ABD$ , has area 64, and its height is $AD = 8 \sqrt{2}$ ; multiply one third by their product to get the volume.

Tetra_1

In the above diagram, a tetrahedron - a triangular pyramid - with vertices is shown inside a cube. Give the volume of the tetrahedron.

Statement 1: The perimeter of Square is 16.

Statement 2: The area of $\bigtriangleup EGH$ is 8.

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.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation

The volume of the pyramid is one third the product of height and the area of its base, which in turn, since here it is a right triangle, is half the product of the lengths and of its legs. Since the prism in the figure is a cube, the three lengths are equal, so we can set each to . The volume of the pyramid is

$V = \frac{1}{3} \cdot s\cdot \frac{1}{2} s\cdot s$

$V = \frac{1}{6} s^{3}$

Therefore, knowing the length of one edge of the cube is sufficient to determine the volume of the pyramid.

Assume Statement 1 alone. Since the perimeter of Square is 16, each side of the square, and each edge of the cube has one fourth this measure, or 4.

Assume Statement 2 alone. $\bigtriangleup EGH$ has congruent legs, each of measure ; since its area is 8, can be found as follows:

$\frac{1}{2} s^{2} = A$

$\frac{1}{2} s^{2} = 8$

$s^{2}= 16$

From either statement alone, the length of each side of the cube, and, subsequently, the volume of the pyramid, can be calculated.

What is the area of a parallelogram with four equal sides?

(1) Each side is $\dpi{100} \small 5cm$

(2) One diagonal is $\dpi{100} \small 6cm$

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

$\dpi{100} \small S= \frac{1}{2}\times L1\times L2$ ( $\dpi{100} \small L1$ and $\dpi{100} \small L2$ are the diagonals of the rhombus). From statement (1) we cannot get to the lengths of diagonals. From statement (2) we only know the length of one diagonal, which is insufficient. However, putting the two statements together, we can use the Pythagorean Theorem to calculate the other diagonal, and then use the formula to calculate the area.

Tetra_3

Note: Figure NOT drawn to scale.

The above figure shows a rectangular prism with an inscribed tetrahedron, or triangular pyramid, with vertices . What is the volume of the tetrahedron?

Statement 1:

Statement 2:

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.

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.

Explanation

We show that the two statements together provide insufficient information by assuming them both to be true.

The volume of the pyramid is one third the product of the height, which is , and the area of the base; this base, being a right triangle, is equal to one half the product of the lengths of its legs, or and . Therefore,

$V = \frac{1}{3} \cdot AE \cdot \frac{1}{2} EF \cdot EH$

$V = \frac{1}{6} \cdot AE \cdot EF \cdot EH$

By Statement 1, , and by Statement 2, , so by substitution,

$V = \frac{1}{6} \cdot AE \cdot AE \cdot 40$

$V = \frac{20}{3} \cdot (AE )^{2}$

Without any further information, however, the volume cannot be determined.

A regular tetrahedron is a solid with four faces, each of which is an equilateral triangle.

Give the volume of a regular tetrahedron.

Statement 1: Each edge has length 8.

Statement 2: Each face has area $16 \sqrt{3}$ .

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.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation

The formula for the volume of a regular tetrahedron given the length of each edge is

$V = \frac{s^{3} }{6\sqrt{2}}$ .

Statement 1 gives information explicitly. Statement 2 gives the means to find , since, if $16 \sqrt{3}$ is substituted for in the formula for an equilateral triangle:

$\frac{s^{2}\sqrt{3}}{4}= A$ ,

the value of can be determined.

Tetra_1

In the above diagram, a tetrahedron—a triangular pyramid—with vertices is shown inside a cube. Give the volume of the tetrahedron.

Statement 1: The cube can be inscribed inside a sphere with volume $972 \pi$ .

Statement 2: A sphere with surface area $108 \pi$ can be inscribed inside the cube.

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.

BOTH statements TOGETHER are insufficient 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.

Explanation

$V = \frac{1}{3} \cdot s\cdot \frac{1}{2} s\cdot s$

$V = \frac{1}{6} s^{3}$

Therefore, knowing the length of one edge of the cube is sufficient to determine the volume of the pyramid.

Assume Statement 1 alone. If the volume of the circumscribing sphere is known to be $972 \pi$ , the radius can be calculated as follows:

$\frac{4}{3} \pi r^{3} = V$

$\frac{4}{3} \pi r^{3} =972 \pi$

$\frac{3}{4\pi} \cdot \frac{4}{3} \pi r^{3} =\frac{3}{4\pi} \cdot 972 \pi$

$r^{3} =729$

$r = \sqrt[3]{729}= 9$

The diameter, which is twice this, or 18, is the length of a diagonal of the cube. By the three-dimensional extension of the Pythagorean Theorem, the relationship of this length to the side length of the cube is

$s^{2}+s^{2}+s^{2} = d^{2}$ , or

$3s^{2} = d^{2}$

$3s^{2} = 18^{2}$

$s^{2} = \frac{18^{2}}{3} = 108$

$s = \sqrt{108} = 6 \sqrt{3}$ ,

Assume Statement 2 alone. If the surface area of the inscribed sphere is known to be $108 \pi$ , then its radius can be calculated as follows:

$4 \pi r^{2} = A$

$4 \pi r^{2} = 108 \pi$

$\frac{4 \pi r^{2} }{4 \pi }=\frac{ 108 \pi}{4 \pi }$

$r^{2} = 27$

$r = \sqrt{27} = 3\sqrt{3}$ .

The diameter of the inscribed sphere, which is twice this, or $6 \sqrt{3}$ , is equal to the length of one edge of the cube.

Either statement alone gives us the length of one side of the cube, which is enough to allow the volume of the pyramid to be calculated.

Find the length of the edge of a tetrahedron.

Statement 1: The volume is 6.

Statement 2: The surface area is 6.

$\textup{EACH statement ALONE is sufficient.}$

$\textup{Statement 1) ALONE is sufficient, but Statement 2) ALONE is not sufficient to answer the question.}$

$\textup{Statement 2) ALONE is sufficient, but Statement 1) ALONE is not sufficient to answer the question.}$

$\textup{BOTH statements taken TOGETHER are sufficient to answer the question, but neither statement ALONE is sufficient.}$

$\textup{BOTH statements TOGETHER are NOT sufficient, and additional data is needed to answer the question. }$

Explanation

Statement 1:) The volume is 6.

Write the formula to find the edge of the tetrahedron given the volume.

$V=\frac{e^3}{6\sqrt2}$

Given the volume, it is possible to find the edge of the tetrahedron.

Statement 2:) The surface area is 6.

Write the formula to find the edge of the tetrahedron given the surface.

$SA=\sqrt3 e^2$

Substitute the surface area to find the edge.

Therefore:

$\textup{EACH statement ALONE is sufficient.}$

A tetrahedron is a solid with four triangular faces.

Give the volume of a tetrahedron.

Statement 1: The tetrahedron has four equilateral faces.

Statement 2: The surface area of the tetrahedron is $120 \sqrt{3}$ .

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.

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 insufficient to answer the question.

Explanation

Neither statement is enough to determine the volume of the tetrahedron; Statement 1 alone gives no actual measurements, and Statement 2 gives only the surface area, which can apply to infinitely many tetrahedrons.

Assume both statements to be true. A tetrahedron with four equilateral faces is a regular tetrahedron, whose surface area, relative to the common length of its edges, is defined by the formula

$A = s^{2}\sqrt{3}$ .

By substituting $120 \sqrt{3}$ for , it is possible to calculate . Consequently, the volume of the tetrahedron can be calculated using the volume formula

$V = \frac{s^{3} }{6\sqrt{2}}$ .

What is the value of $\dpi{100} \small 2x+2y$ ?

Statement 1: $\dpi{100} \small x-3y=4$

Statement 2: $\dpi{100} \small x+y=4$

Statement 2 ALONE is sufficient, but statement 1 is not sufficient.

Statement 1 ALONE is sufficient, but statement 2 is not sufficient.

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

EACH statement ALONE is sufficient.

Statements 1 and 2 TOGETHER are NOT sufficient.

Explanation

We know that we need 2 equations to solve for 2 variables, so it is tempting to say that both statements are needed. This is actually wrong! We aren't being asked for the individual values of x and y, instead we are being asked for the value of an expression.

$\dpi{100} \small 2x+2y$ is just $\dpi{100} \small 2\left ( x+y \right )$ , and statement 2 gives us the value of $\dpi{100} \small x+y$ . For data sufficiency questions, we don't actually have to solve the question, but if we wanted to, we would simply multiply statement 2 by 2.

$\dpi{100} \small 2x+2y=2\left ( x+y \right )=2$ * Statement 2 = $\dpi{100} \small 2\times 4=8$

$\bigtriangleup ABC$ is a right triangle. Evaluate .

Statement 1:

Statement 2:

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.

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.

Explanation

Neither statement alone gives enough information to find , as each alone gives only one sidelength.

Assume both statements are true. While neither side is indicated to be a leg or the hypotenuse, the hypotenuse of a right triangle is longer than either leg; therefore, since $\overline {AC}$ and $\overline {BC}$ are of equal length, they are the legs. $\overline {AB}$ is the hypotenuse of an isosceles right triangle with legs of length 10, and by the 45-45-90 Theorem, the length of $\overline{AB}$ is $\sqrt{2}$ times that of a leg, or $10 \sqrt{2}$ .

Page 1 of 100

Return to subject

AI TutorYour personal study buddy

Question of the DayDaily practice to build your skills

FlashcardsStudy and memorize key concepts

WorksheetsBuild custom practice quizzes

AI SolverStep-by-step problem solutions

Learn by ConceptMaster concepts step by step

Diagnostic TestsAssess your knowledge level

Practice TestsTest your skills with exam questions

GamesLearn through free games