GMAT Math : GMAT Quantitative Reasoning

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #1 : Dsq: Calculating The Length Of The Side Of An Equilateral Triangle

Given equilateral triangle \(\displaystyle \bigtriangleup ABC\) and right triangle \(\displaystyle \bigtriangleup DEF\), which, if either, is longer,  \(\displaystyle \overline{AB}\) or \(\displaystyle \overline{DE}\) ?

Statement 1: \(\displaystyle \overline{BC} \cong \overline{EF}\)

Statement 2: \(\displaystyle \angle D\) is a right angle.

Possible Answers:

EITHER statement ALONE is 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 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:

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

Explanation:

Assume Statement 1 alone. Since all three sides of \(\displaystyle \bigtriangleup ABC\) are congruent - specifically, \(\displaystyle \overline{AB} \cong \overline{BC}\) - and \(\displaystyle \overline{BC} \cong \overline{EF}\), it follows by transitivity that \(\displaystyle \overline{AB} \cong \overline{EF}\). However, no information is given as to whether \(\displaystyle \overline{DE}\) has length greater then, equal to, or less than \(\displaystyle \overline{EF}\), so which of \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{DE}\), if either, is the longer cannot be answered.

Assume Statement 2 alone. Since \(\displaystyle \angle D\) is the right angle of \(\displaystyle \bigtriangleup DEF\)\(\displaystyle \overline{EF}\) is the hypotenuse and this the longest side, so \(\displaystyle EF > DE\) and \(\displaystyle EF > DF\). However, no comparisons with the sides of \(\displaystyle \bigtriangleup ABC\) can be made.

Now assume both statements are true. \(\displaystyle AB = EF\) as a consequence of Statement 1, and \(\displaystyle EF > DE\) as a consequence of Statement 2, so \(\displaystyle AB > DE\).

Example Question #4 : Equilateral Triangles

What is the length of side \(\displaystyle \overline{AB}\) of equilateral triangle \(\displaystyle \bigtriangleup ABC\) ?

Statement 1: \(\displaystyle \overline{BC}\) is a diagonal of Rectangle \(\displaystyle BXCY\) with area 30.

Statement 2: \(\displaystyle \overline{AC}\) is a diagonal of Square \(\displaystyle AMCN\) with area 36.

Possible Answers:

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.

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.

Correct answer:

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

Explanation:

An equilateral triangle has three sides of equal measure, so if the length of any one of the three sides can be determined, the lengths of all three can be as well.

Assume Statement 1 alone. \(\displaystyle \overline{BC}\) is a diagonal of a rectangle of area 30. However, neither the length nor the width can be determined, so the length of this segment cannot be determined with certainty.

Assume Statement 2 alone. A square with area 36 has sidelength the square root of this, or 6; its diagonal, which is \(\displaystyle \overline{AC}\), has length \(\displaystyle \sqrt{2}\) times this, or \(\displaystyle 6 \sqrt{2}\). This is also the length of \(\displaystyle \overline{AB}\).

 

Example Question #5 : Dsq: Calculating The Length Of The Side Of An Equilateral Triangle

What is the length of side \(\displaystyle \overline{BC}\) of equilateral triangle \(\displaystyle \bigtriangleup ABC\) ?

Statement 1: \(\displaystyle A\)\(\displaystyle B\), and \(\displaystyle C\) are all located on a circle with area \(\displaystyle 576 \pi\).

Statement 2: The midpoints of all three sides are located on a circle with circumference \(\displaystyle 24 \pi\).

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 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:

We demonstrate that either statement alone yields sufficient information by noting that the circle that includes all three vertices of a triangle - described in Statement 1 - is its circumscribed circle, and that the circle that includes all three midpoints of the sides of an equilateral triangle - described in Statement 2 - is its inscribed circle. We examine this figure below, which shows the triangle, both circles, and the three altitudes:

Thingy_5

The three altitudes intersect at \(\displaystyle O\), which divides each altitude into two segments whose lengths have ratio 2:1. \(\displaystyle O\) is the center of both the circumscribed circle, whose radius is \(\displaystyle OC\), and the inscribed circle, whose radius is \(\displaystyle OM\).

Therefore, from Statement 1 alone and the area formula for a circle, we can find \(\displaystyle OC\) from the area \(\displaystyle 576\pi\) of the circumscribed circle:

\(\displaystyle \pi (OC)^{2} =a\)

\(\displaystyle \pi (OC)^{2} = 576 \pi\)

\(\displaystyle (OC)^{2} =576\)

\(\displaystyle OC= 24\)

From Statement 2 alone and the circumference formula for a cicle, we can find \(\displaystyle OM\) from the circumference \(\displaystyle 24\pi\) of the inscribed circle:

\(\displaystyle 2 \pi \cdot OM = c\)

\(\displaystyle 2 \pi \cdot OM = 24 \pi\)

\(\displaystyle OM =12\)

By symmetry, \(\displaystyle \bigtriangleup COM\) is a 30-60-90 triangle, and either way, \(\displaystyle CM = 12\sqrt{3}\), and \(\displaystyle BC =2 \cdot CM =2 \cdot 12\sqrt{3}= 24 \sqrt{3}\).

Example Question #5 : Dsq: Calculating The Length Of The Side Of An Equilateral Triangle

Given two equilateral triangles \(\displaystyle \bigtriangleup ABC\) and \(\displaystyle \bigtriangleup DEF\), which, if either, is greater, \(\displaystyle AB\) or \(\displaystyle DE\) ?

Statement 1: \(\displaystyle AC> EF\)

Statement 2: \(\displaystyle BC = DF + 1\)

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 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.

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

An equilateral triangle has three sides of equal length, so \(\displaystyle AB = AC = BC\) and \(\displaystyle DE = EF = DF\).

Assume Statement 1 alone. Since \(\displaystyle AC> EF\), then, by substitution, \(\displaystyle AB > DE\).

Assume Statement 2 alone. Since \(\displaystyle BC = DF + 1\), it follows that \(\displaystyle BC > DF\), and again by substitution, \(\displaystyle AB > DE\).

Example Question #7 : Dsq: Calculating The Length Of The Side Of An Equilateral Triangle

You are given two equilateral triangles \(\displaystyle \bigtriangleup ABC\) and \(\displaystyle \bigtriangleup DEF\).

Which, if either, is greater, \(\displaystyle AB\) or \(\displaystyle DE\) ?

Statement 1: The perimeters of \(\displaystyle \bigtriangleup DEF\) and \(\displaystyle \bigtriangleup ABC\) are equal.

Statement 2: The areas of \(\displaystyle \bigtriangleup DEF\) and \(\displaystyle \bigtriangleup ABC\)are equal.

Possible Answers:

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. 

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:

Assume Statement 1 alone, and let \(\displaystyle P\) be the common perimeter of the triangles. Since an equilateral triangle has three sides of equal length, \(\displaystyle AB = \frac{1}{3} P\) and \(\displaystyle DE = \frac{1}{3} P\), so \(\displaystyle AB = DE\).

Assume Statement 2 alone, and let \(\displaystyle A\) be the common area of the triangles. Using the area formula for an equilateral triangle, we can note that:

\(\displaystyle A = \frac{s^{2}\sqrt{3}}{4}\)

\(\displaystyle A = \frac{ (AB)^{2}\sqrt{3}}{4}\) and \(\displaystyle A = \frac{ (DE)^{2}\sqrt{3}}{4}\),  

so

\(\displaystyle \frac{ (AB)^{2}\sqrt{3}}{4}= \frac{ (DE)^{2}\sqrt{3}}{4}\)

\(\displaystyle \frac{ (AB)^{2}\sqrt{3}}{4}\cdot \frac{4}{\sqrt{3}}= \frac{ (DE)^{2}\sqrt{3}}{4} \cdot \frac{4}{\sqrt{3}}\)

\(\displaystyle (AB)^{2} = (DE)^{2}\)

\(\displaystyle AB = DE\).

Example Question #11 : Dsq: Calculating The Length Of The Side Of An Equilateral Triangle

Given equilateral triangle \(\displaystyle \bigtriangleup ABC\) and right triangle \(\displaystyle \bigtriangleup DEF\), which, if either, is longer,  \(\displaystyle \overline{AB}\) or \(\displaystyle \overline{DE}\) ?

Statement 1: \(\displaystyle \overline{BC} \cong \overline{EF}\)

Statement 2: \(\displaystyle \overline{AC} \cong \overline{DF}\)

Possible Answers:

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.

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:

Assume Statement 1 alone. Since all three sides of \(\displaystyle \bigtriangleup ABC\) are congruent - specifically, \(\displaystyle \overline{AB} \cong \overline{BC}\) - and \(\displaystyle \overline{BC} \cong \overline{EF}\), it follows by transitivity that \(\displaystyle \overline{AB} \cong \overline{EF}\). However, no information is given as to whether \(\displaystyle \overline{DE}\) has length greater than, equal to, or less than \(\displaystyle \overline{EF}\), so it cannot be determined which of \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{DE}\), if either, is the longer. By a similar argument, Statement 2 yields insufficient information.

Now assume both statements are true. \(\displaystyle \overline{DF}\) and \(\displaystyle \overline{EF}\) are each congruent to one of the congruent sides of equilateral \(\displaystyle \bigtriangleup ABC\) and are therefore congruent to each other. However, the hypotenuse of a right triangle must be longer than both legs, so the hypotenuse of  \(\displaystyle \bigtriangleup DEF\) is \(\displaystyle \overline{DE}\)\(\displaystyle \overline{DE}\) is also longer than any segment congruent to one of the legs, which includes all three sides of \(\displaystyle \bigtriangleup ABC\) - specificially, \(\displaystyle \overline{DE}\) is longer than \(\displaystyle \overline{AB}\).

Example Question #12 : Equilateral Triangles

\(\displaystyle \bigtriangleup ABC\) is equilateral. \(\displaystyle \bigtriangleup DEF\) may or may not be equilateral. 

which, if either, is longer,  \(\displaystyle \overline{AB}\) or \(\displaystyle \overline{DE}\) ?

Statement 1: \(\displaystyle BC= DF + EF\)

Statement 2: \(\displaystyle AC> DF\) and \(\displaystyle AC> EF\)

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:

Assume Statement 1 alone. \(\displaystyle \bigtriangleup ABC\) is equilateral, so \(\displaystyle AB = BC\). Also, by the Triangle Inequality, the sum of the lengths of two sides of a triangle must exceed the third, so \(\displaystyle DF + EF > DE\). From Statement 1, \(\displaystyle BC= DF + EF\), so by substitution, \(\displaystyle BC > DE\), and \(\displaystyle AB> DE\).

Statement 2 alone provides insufficient information. For example, assume \(\displaystyle \bigtriangleup ABC\) is an equilateral triangle with sidelength 9. If \(\displaystyle \bigtriangleup DEF\) is an equilateral triangle with sidelength 8, the conditions of the statement hold, and \(\displaystyle AB > DE\). However, if \(\displaystyle \bigtriangleup DEF\) is a right triangle in which \(\displaystyle DF = 6\)\(\displaystyle EF = 8\), and \(\displaystyle DE = 10\), the conditions of the statement still hold, but \(\displaystyle AB < DE\).

Example Question #13 : Equilateral Triangles

Given equilateral triangles \(\displaystyle \bigtriangleup ABC\) and \(\displaystyle \bigtriangleup DEF\), which, if either, is longer,  \(\displaystyle \overline{AB}\) or \(\displaystyle \overline{DE}\) ?

Statement 1: \(\displaystyle AC + DF = 15\)

Statement 2: \(\displaystyle BC \cdot EF = 50\)

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. 

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question. 

Explanation:

All sides of an equilateral triangle have the same measure, so we can let \(\displaystyle x\) be the common sidelength of \(\displaystyle \bigtriangleup ABC\), and \(\displaystyle y\) be that of \(\displaystyle \bigtriangleup DEF\).

Statement 1 can be rewritten as \(\displaystyle x+ y = 15\); Statement 2 can be rewritten as \(\displaystyle xy = 50\). The equivalent question is whether we can determine which, if either, is greater, \(\displaystyle x\) or \(\displaystyle y\:\). The two statements together are insufficient to answer the question, however; 5 and 10 have sum 15 and product 50, but we cannot determine without further information whether  \(\displaystyle x = 5\) and \(\displaystyle y = 10\), or vice versa. Therefore, we do not know for sure whether a side of \(\displaystyle \bigtriangleup ABC\) is longer than a side of \(\displaystyle \bigtriangleup DEF\) - specifically, which of \(\displaystyle \overline{AB}\) or \(\displaystyle \overline{DE}\) is longer.

Example Question #12 : Dsq: Calculating The Length Of The Side Of An Equilateral Triangle

Given equilateral triangles \(\displaystyle \bigtriangleup ABC\) and \(\displaystyle \bigtriangleup DEF\), which, if either, is longer,  \(\displaystyle \overline{AB}\) or \(\displaystyle \overline{DE}\) ?

Statement 1: \(\displaystyle BC+ EF= 24\)

Statement 2: \(\displaystyle AC \cdot DF = 144\)

Possible Answers:

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. 

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:

All sides of an equilateral triangle have the same measure, so we can let \(\displaystyle x\) be the common sidelength of \(\displaystyle \bigtriangleup ABC\), and \(\displaystyle y\:\) be that of \(\displaystyle \bigtriangleup DEF\).

Statement 1 can be rewritten as \(\displaystyle x+ y = 24\); Statement 2 can be rewritten as \(\displaystyle xy = 144\). The equivalent question is whether we can determine which, if either, is greater, \(\displaystyle x\) or \(\displaystyle y\:\).

Statement 1 alone yields insufficient information; for example, the two numbers added together could be 10 and 14, but it is impossible to determine whether \(\displaystyle x\) or \(\displaystyle y\) is the greater of the two. Statement 2 alone is also insufficient, for a similar reason; for example, the two numbers could be 9 and 16, but again, either \(\displaystyle x\) or \(\displaystyle y\) could be the greater.

Now assume both statements. The only two numbers that can be added to yield a sum of 24 and multiplied to yield a product of 144 are 12 and 12; therefore, \(\displaystyle x=y\), and \(\displaystyle \bigtriangleup ABC\) and \(\displaystyle \bigtriangleup DEF\) have the same sidelengths. Specifically, \(\displaystyle \overline{AB}\) and \(\displaystyle \overline{DE}\) have the same length.

Example Question #1 : Dsq: Calculating The Area Of An Equilateral Triangle

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What is the area of \(\displaystyle \bigtriangleup ABC\)?

(1) The height \(\displaystyle \overline{CD}\) is 5.

(2) The base \(\displaystyle \overline{AB}\) is 4.

Possible Answers:

Statement 2 alone is sufficient

Each statement alone is sufficient

Statements 1 and 2 together are not sufficient

Both statements taken together are sufficient

Statement 1 alone is sufficient

Correct answer:

Both statements taken together are sufficient

Explanation:

To find an area of a triangle we need the length of the height and the length of the corresponding basis.

Each statement 1 and 2 alone is not sufficient, since we don't know whether the triangle is equilateral. Indeed, we need to take both statements to be able to calculate the area.

Hence, both statements together are sufficient.

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