GMAT Math : Data-Sufficiency Questions

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Example Questions

Example Question #1 : Dsq: Calculating The Percentage Of A Sector From An Angle

Dirk bought a sticker to cover part of the base of his scuba tank. The sticker and the base have the same radius. Given the following, find the percentage of the base that the sticker covers.

I) The central angle of the sticker is $66^{\circ}$

II) The base of the tank has a radius of 5 inches.

Possible Answers:

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

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

Either statement is 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.

Correct answer:

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

Explanation:

To find the percentage that the sticker covers, we need to know what portion of the circle it covers. We could find this either with the area of the circle or the area of the sector, but we don't have a good way to find the area of the sector.

Statement I gives us the degree measurement of the sticker. We can make a fraction, $\frac{66}{360}$ , to find the percentage of the base covered.

Statement II gives us the radius of the base, which lets us also know the radius of the sticker; however, this will not be helpful in finding the percentage of the base covered, so Statement II is not really helpful.

Thus, Statement I is sufficient and Statement II is not.

Recap:

Dirk bought a sticker to cover part of the base of his scuba tank. The sticker and the base have the same radius. Given the following, find the percentage of the base that the sticker covers.

I) The central angle of the sticker is $66^{\circ}$

II) The base of the tank has a radius of 5 inches

So, according to Statement I, the angle of the sticker is 66 degrees. Because we are dealing with a circle, we can find the percentage of the circle covered by putting 66 over 360.

$\frac{66}{360}=.1833\bar{3}$

The sticker covers $18\frac{1}{3} \%$ of the scuba tank.

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Example Question #2 : Dsq: Calculating The Percentage Of A Sector From An Angle

Gmat circle

What is the area of sector AOB?

1) $\angle ACB=30 degrees$

2) Arclength of AB is 5in

Possible Answers:

Statement 2 alone is sufficient.

Either of the statements is sufficient.

Neither of the statements, separate or together, is sufficient.

Statement 1 alone is sufficient.

Together the two statements are sufficient.

Correct answer:

Together the two statements are sufficient.

Explanation:

The angle $\angle AOB = 2\angle ACB$ , so $\angle AOB = 60$ .

The Arc length can be related to the radius as follows:

$AB = 2\pi r (\frac{\angle AOB}{360})$

$r= \frac{5in}{2\pi}(\frac{360}{60})=\frac{15}{\pi}in$

From there, the area of the sector can be found:

$AOB=\pi r^2(\frac{\angle AOB}{360})=\pi(\frac{15}{\pi})^2(\frac{1}{6})in^2$

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Example Question #591 : Data Sufficiency Questions

Consider circle F with sector BFG. What percent of F is BFG?

I) Circle F has a radius of 5 meters and a circumference of $10\pi$ meters.

II) Sector BFG has a central angle measure of $315^{\circ}$ .

Possible Answers:

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

Either statement is sufficient to answer the question.

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.

Correct answer:

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

Explanation:

Consider circle F with sector BFG. What percent of F is BFG?

I) Circle F has a radius of 5 meters and a circumference of $10\pi$ meters

II) Sector BFG has a central angle measure of $315^{\circ}$

To find the percentage of a circle covered by a sector, we need to know the central angle. A circle will always have $360^{\circ}$ , so if we put the angle given to us over $360^{\circ}$ . So, II) allows us to answer the question:

$\frac{315}{360}\ast100=87.5%$

So sector BFG is 87.5% of circle F

I) Give us interesting information, but not helpful in regards to the current question. It doesn't matter what the radius is, and the circumference is also irrelevant.

Therefore, II) is sufficient, but I) is not.

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Example Question #592 : Data Sufficiency Questions

Jane wants to put a sector-shaped sticker on the bottom of her cylindrical water bottle. The water bottle has a circular base. Find the percent of the base that the sticker covers

I) The sticker has a thickness of 0.05mm .

II) The central angle of the sticker has a measure of $\frac{5\pi}{6}$ .

Possible Answers:

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.

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.

Both statements are needed to answer the question.

Correct answer:

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

Explanation:

Jane wants to put a sector-shaped sticker on the bottom of her cylindrical water bottle. The water bottle has a circular base. Find the percent of the base that the sticker covers

I) The sticker has a thickness of

II) The central angle of the sticker has a measure of $\frac{5\pi}{6}$

I) is irrelevant. The thickness of the sticker doesn't help us find the area it covers. Statment I) is trying to distract you with 3 dimensions, when we only need to worry about 2.

II) is much more helpful. If we know the central angle, we can find the percentage the sector covers. To find the "answer" perform the following:

$\frac{\frac{5\pi}{6}}{2\pi}=\frac{5\pi}{6}\ast\frac{1}{2\pi}=\frac{5}{12}$

So we are dealing with $\frac{5}{12}$ of the circle, or 41.67%

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Example Question #591 : Data Sufficiency Questions

Weird

The above figure shows two quarter circles inscribed inside a rectangle. What is the total area of the white region?

Statement 1: The area of the black region is $50\pi$ square centimeters.

Statement 2: The rectangle has perimeter 60 centimeters.

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.

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 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 width of the rectangle is equal to the radius of the quarter circles, which we call ; the length is twice that, or .

The area of the rectangle is $r \cdot 2r = 2r^{2}$ ; the total area of the two black quarter circles is $2 \cdot \frac{1}{4} \cdot \pi r^{2} = \frac{1}{2} \pi r^{2}$ , so the area of the white region is their difference,

$2r^{2} - \frac{1}{2} \pi r^{2}$

Therefore, all that is needed to find the area of the white region is the radius of the quarter circle.

If we know that the area of the black region is $50 \pi$ centimeters, then we can deduce using this equation:

$\frac{1}{2} \pi r^{2} = 50 \pi$

If we know that the perimeter of the rectangle is 60 centimeters, we can deduce via the perimeter formula:

2 r + 2 (2r) = 60

Either statement alone allows us to find the radius and, consequently, the area of the white region.

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Example Question #2 : Dsq: Calculating The Area Of A Sector

Sector

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: The circle has circumference $40 \pi$ .

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

Possible Answers:

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.

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.

Correct answer:

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

Explanation:

To find the area of a sector of a circle, we need a way to find the area of the circle and a way to find the central angle of the sector.

Statement 1 alone gives us the circumference; this can be divided by $2 \pi$ to yield the radius, and that can be substituted for in the formula $A = \pi r^{2}$ to find the area. However, it provides no clue that might yield $m \angle AOB$ .

Statement 2 alone asserts that $m \angle ACB = 30 ^{\circ }$ . This is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle that intercepts it - has twice this measure, or $60 ^{\circ }$ . Therefore, Statement 2 alone gives the central angle, but does not yield any clues about the area.

Assume both statements are true. The radius is $40 \pi \div 2 \pi = 20$ and the area is $A = \pi \cdot 20^{2 }= 400 \pi$ . The shaded sector is $\frac{60^{\circ }}{360^{\circ }}= \frac{1}{6}$ of the circle, so the area can be calculated to be $\frac{1}{6} \times 400 \pi = \frac{200 \pi}{3}$ .

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Example Question #3 : Dsq: Calculating The Area Of A Sector

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

Statement 2: $m \angle BOD = 120 ^{\circ }$

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.

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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Statement 1 alone asserts that $m \angle ACB = 30 ^{\circ }$ . This is an inscribed angle that intercepts the arc $\widehat{AB}$ ; therefore, the arc - and the central angle $\angle AOB$ that intercepts it - has twice this measure, or $60 ^{\circ }$ .

Statement 2 alone asserts that $m \angle BOD = 120 ^{\circ }$ . By angle addition, $m\angle AOB = 180 ^{\circ } - m \angle BOD = 180 ^{\circ } - 120 ^{\circ } = 60 ^{\circ }$ .

Either statement alone tells us that the shaded sector is $\frac{60^{\circ }}{360^{\circ }}= \frac{1}{6}$ of the circle, and that the white sector is $\frac{5}{6}$ of it; it can be subsequently calculated that the ratio of the areas is $\frac{5}{6} : \frac{1}{6}$ , or 5:1 .

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Example Question #4 : Dsq: Calculating The Area Of A Sector

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: AD = 100

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

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.

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.

EITHER statement ALONE is 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:

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 2:1 .

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Example Question #1 : Dsq: Calculating The Area Of A Sector

Sector

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: $m \angle BCD = 120^{\circ }$ .

Statement 2: The circle has circumference $28 \pi$ .

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.

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.

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:

To find the area of a sector of a circle, we need a way to find the area of the circle and a way to find the central angle $\angle BOD$ of the sector.

Statement 1 alone gives us the circumference; this can be divided by $2 \pi$ to yield radius $28 \pi \div 2 \pi = 14$ , and that can be substituted for in the formula $A = \pi r^{2}$ to find the area: $A = \pi \cdot 14^{2} = 196 \pi$ .

However, it provides no clue that might yield $m \angle BOD$ .

From Statement 2 alone, we can find $m \angle BOD$ . $\angle BCD$ , an inscribed angle, intercepts an arc twice its measure - this arc is $\widehat{BAD}$ , which has measure $240^{\circ }$ . $\widehat{B D}$ , the corresponding minor arc, will have measure $360 ^{\circ}- m \widehat{BAD} = 360 ^{\circ} - 240 ^{\circ} = 120 ^{\circ}$ . This gives us $m \angle BOD$ , but no clue that yields the area.

Now assume both statements are true. The area is $196 \pi$ and the shaded sector is $\frac{120^{\circ }}{360^{\circ }}= \frac{1}{3}$ of the circle, so the area can be calculated to be $\frac{1}{3} \times 196 \pi = \frac{196 \pi}{3}$ .

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Example Question #3 : Dsq: Calculating The Area Of A Sector

Sector

The circle in the above diagram has center . Give the area of the shaded sector.

Statement 1: The sector with central angle $\angle BOD$ has area $48 \pi$ .

Statement 2: $\widehat{BD }= 8 \pi$ .

Possible Answers:

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.

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.

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. No clues are given about the measure of $\angle BOD$ , so that of $\angle AOB$ , and, subsequently, the area of the shaded sector, cannot be determined.

Assume Statement 2 alone. Since the circumference of the circle is not given, it cannot be determined what part of the circle $\widehat{BD }$ , or, subsequently, $\widehat{AB}$ , is, and therefore, the central angle of the sector cannot be determined. Also, no information about the area of the circle can be determined.

Now assume both statements are true. Let be the radius of the circle and $N ^{\circ }$ be the measure of $\angle BOD$ . Then:

$\frac{N}{360} \cdot \pi r ^{2} = 48 \pi$

and

$\frac{N}{360} \cdot 2 \pi r = 8 \pi$

The statements can be simplified as

$r ^{2}N = 17,280$

and

r N= 1,440

From these two statements:

$\frac{r ^{2}N }{rN}=\frac{ 17,280}{1,440}$

r = 12 ; the second statement can be solved for :

12 N= 1,440

N = 120 .

$m \angle BOD = 120^{\circ }$ , so $m \angle AOB = 60^{\circ }$ .

Since r = 12 , the circle has area $A = \pi r^{2} = \pi \cdot 12^{2} = 144 \pi$ . Since we know the central angle of the shaded sector as well as the area of the circle, we can calculate the area of the sector as

$\frac{1}{6} \times 144 \pi = 24\pi$ .

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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 #1 : Dsq: Calculating The Percentage Of A Sector From An Angle

Example Question #2 : Dsq: Calculating The Percentage Of A Sector From An Angle

Example Question #591 : Data Sufficiency Questions

Example Question #592 : Data Sufficiency Questions

Example Question #591 : Data Sufficiency Questions

Example Question #2 : Dsq: Calculating The Area Of A Sector

Example Question #3 : Dsq: Calculating The Area Of A Sector

Example Question #4 : Dsq: Calculating The Area Of A Sector

Example Question #1 : Dsq: Calculating The Area Of A Sector

Example Question #3 : Dsq: Calculating The Area Of A Sector

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