GMAT Math : DSQ: Calculating the angle of a sector

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

What time is it?

Statement 1: The minute hand and the hour hand are currently forming a $45^{\circ }$ angle.

Statement 2: The tip of the minute hand has traveled exactly eight inches since last leaving the 12 position.

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:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

The two statements together are not enough unless you know the size of the minute hand; without this information, you cannot tell the angular position of the minute hand, so, even if you know the angle the hands are making, you do not know the position of the hour hand either.

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

What time is it?

Statement 1: The minute hand and the hour hand form a $70 ^{\circ }$ angle.

Statement 2: The minute hand is exactly on the 8.

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.

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.

Correct answer:

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

Explanation:

The first event happens numerous times over the course of twelve hours, so the first statement is not enough to deduce the time; all the second statement tells you is that it is forty minutes after an hour (12:40, 1:40, etc.)

Suppose we put the two statements together. It is $360 \div 12 = 30 ^{\circ }$ from one number to the next, so the 8:00 position is the $30 \cdot 8 = 240 ^{\circ}$ position. If the hour hand makes a $70 ^{\circ }$ with the minute hand, then the hour hand is either at $240 -70 = 170 ^{\circ }$ or $240 +70 = 310 ^{\circ }$ . Since forty minutes is two-thirds of an hour, however, the hour hand must be two-thirds of the way from one number to the next.

Case 1: If the hour hand is at $170 ^{\circ }$ , then it is at the $\frac{170 }{30} = 5 \frac{2}{3}$ position - in other words, two-thirds of the way from the 5 to the 6. This is consistent with our conditions.

Case 2: If the hour hand is at $310 ^{\circ }$ , then it is at the $\frac{310 }{30} = 10 \frac{1}{3}$ position - in other words, one-third of the way from the 10 to the 11. This is inconsistent with our conditions.

Therefore, only the first case is possible, and if we are given both statements, we know it is 5:40.

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

What time is it?

Statement 1: The minute hand and the hour hand are currently forming a $75^{\circ }$ angle.

Statement 2: The minute hand is on the 6.

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Since there are twelve numbers on the clock, the angular measure from one number to the next is $30^{\circ }$ ; this means $75 ^{\circ }$ represents two and a half number positions.

Suppose we know both statements. Since the minute hand is on the 6, the hour hand is either midway between the 3 and the 4, or midway between the 8 and the 9. Both scenarios are possible, as they correspond to 3:30 and 8:30, respectively, so the question is not answered even if we know both statements.

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Example Question #61 : Circles

Chords

Note: Figure NOT drawn to scale.

Refer to the above figure. What is the degree measure of $\widehat{AB }$ ?

Statement 1: $m \angle C = 56^{\circ }$ .

Statement 2: $\widehat{ACB }$ measures $248 ^{\circ }$ .

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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

From Statement 1, the measure of the arc can be determined by doubling the measure of the intercepting angle, which is $\angle C$ . From Statement 2, the measure of the arc can be calculated by subtracting from $360 ^{\circ }$ the degree measure of the corresponding major arc, which is $\widehat{ACB }$ .

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Example Question #5 : Dsq: Calculating The Angle Of A Sector

Election

Above are the results of the election for student body president that Greg is about to publish in the school newspaper. At the last minute, his friend Melissa stops him and reminds him that there was a sixth candidate, Wilson, who got 202 votes.

Greg's article includes a circle graph that he will now have to change to reflect this corrected information. By how many degrees will the angle measure of the sector representing Douglas decrease (nearest whole degree)?

Possible Answers:

$19^{\circ }$

$16^{\circ }$

$10^{\circ }$

$13^{\circ }$

$22^{\circ }$

Correct answer:

$16^{\circ }$

Explanation:

According to Greg's erroneous information, the number of people who voted was

145+213+101+245+176= 880 .

213 voted for Douglas, meaning that his sector will have degree measure

$\frac{213}{880} \times 360^{\circ} \approx 87^{\circ }$

Based on the new information,

880 + 202 = 1,082 people voted.

213 voted for Douglas, meaning that his sector will have degree measure

$\frac{213}{1,082} \times 360^{\circ} \approx 71^{\circ }$

The reduction in degree measure will be $87^{\circ } - 71^{\circ } = 16^{\circ }$ .

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Example Question #6 : Dsq: Calculating The Angle Of A Sector

Election

After a student body election, Henry is constructing a circle graph to represent the above voter count.

What will be the measure of the central angle of the sector representing Thomas (nearest whole number)?

Possible Answers:

$59^{\circ }$

$87^{\circ }$

$41^{\circ }$

$72^{\circ }$

$100^{\circ }$

Correct answer:

$72^{\circ }$

Explanation:

The number of people who voted:

145+213+101+245+176= 880

176 people voted for Starr, so the sector representing Starr will have measure

$\frac{176}{880} \times 360^{\circ } \approx 72^{\circ }$

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Example Question #7 : Dsq: Calculating The Angle Of A Sector

Circle A has radius six times that of Circle B; a sector of Circle A with angle measure $N^{\circ }$ has the same area as Circle B. Evaluate .

Possible Answers:

$30^{\circ }$

$20^{\circ }$

$60^{\circ }$

$5^{\circ }$

$10^{\circ }$

Correct answer:

$10^{\circ }$

Explanation:

Let be the radius of Circle B. Then Circle A has radius and, subsequently, area $\pi \left ( 6r\right )^{2} = 36 \pi r ^{2}$ . Since the area of Circle B is $\pi r^{2}$ , the area of Circle A is 36 times that of Circle B.

The given sector of Circle A has the same area as Circle B, so the sector is one thirty-sixth of the circle. That makes the angle measure of the sector

$\frac{1}{36} \times 360 ^{\circ } = 10^{\circ }$

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Example Question #8 : Dsq: Calculating The Angle Of A Sector

The radius of Circle A is equal to the sidelength of Square B. A sector of Circle A has the same area as Square B. Which of the following is the degree measure of this sector?

Possible Answers:

$\left (\frac{360}{\pi} \right )^{\circ }$

$\left (\frac{60}{\pi} \right )^{\circ }$

$\left (\frac{90}{\pi} \right )^{\circ }$

$\left (\frac{180}{\pi} \right )^{\circ }$

$\left (\frac{120}{\pi} \right )^{\circ }$

Correct answer:

$\left (\frac{360}{\pi} \right )^{\circ }$

Explanation:

The radius of Circle A and the length of a side of the square are the same - we will call each . The area of the circle is $A_{a}= \pi r^{2}$ ; that of the square is $A_{b}= r^{2}$ . Therefore, a sector of the circle with area $r^{2}$ will be $\frac{1}{\pi}$ of the circle, which is a sector of measure

$\frac{1}{\pi} \times 360 ^{\circ } = \left (\frac{360}{\pi} \right )^{\circ }$

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Example Question #9 : Dsq: Calculating The Angle Of A Sector

Election

After a student body election, Henry is constructing a circle graph to represent the above voter count.

What will be the measure of the central angle of the sector representing Starr (nearest whole number)?

Possible Answers:

$41^{\circ }$

$87^{\circ }$

$100^{\circ }$

$59^{\circ }$

$72^{\circ }$

Correct answer:

$100^{\circ }$

Explanation:

The number of people who voted:

145+213+101+245+176= 880

245 people voted for Starr, so the sector representing Starr will have measure

$\frac{245}{880} \times 360^{\circ } \approx 100^{\circ }$

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Example Question #10 : Dsq: Calculating The Angle Of A Sector

Election

Above are the results of the election for student body president that Mike is about to publish in the school newspaper. At the last minute, his friend Veronica stops him and tells him that there was an error in one digit - Lealand got 181 votes, not 101 votes.

Mike's article includes a circle graph that he will now have to change to reflect this corrected information. By how many degrees will the angle measure of the sector representing Lealand increase (nearest whole degree)?

Possible Answers:

$39 ^{\circ }$

$36^{\circ }$

$27^{\circ }$

$33^{\circ }$

$30^{\circ }$

Correct answer:

$27^{\circ }$

Explanation:

According to Mike's erroneous information, the number of people who voted was

145+213+101+245+176= 880 ,

101 of whom voted for Lealand. Therefore, Mike's initial circle graph would have a sector of degree measure

$\frac{101}{880} \times 360^{\circ} = 41.3^{\circ}$

representing Lealand's share of the vote.

However, the corrected figures are

145+213+181+245+176= 960 votes total,

181 of which went to Lealand, so his sector will have measure

$\frac{181}{960} \times 360^{\circ} = 67.9^{\circ}$ ,

an increase of $67.9^{\circ } - 41.3^{\circ }= 26.6^{\circ }$ .

The correct response is $27^{\circ }$ .

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GMAT Math : DSQ: Calculating the angle of a sector

Study concepts, example questions & explanations for GMAT Math

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Example Questions

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

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

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

Example Question #61 : Circles

Example Question #5 : Dsq: Calculating The Angle Of A Sector

Example Question #6 : Dsq: Calculating The Angle Of A Sector

Example Question #7 : Dsq: Calculating The Angle Of A Sector

Example Question #8 : Dsq: Calculating The Angle Of A Sector

Example Question #9 : Dsq: Calculating The Angle Of A Sector

Example Question #10 : Dsq: Calculating The Angle Of A Sector

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