Question: In a circular clock, the long hand is the radius of the circle. At what time is the smaller angle between the hands of the clock NOT divisible by 15?

1. 7:20
2. 9:00
3. 4:30

(A) Only I
(B) Only III
(C) I + II
(D) II+III
(E) I+II+III

Correct Answer: A
Solution and Explanation
Approach Solution 1:
In one minute hand take a 360 Degree circle and the hour hand take a 30 degree circle
For the hour hand 30 degree movement is accomplished in 1 hour (60 Min). So in one minute 30/60 = .5 degree (hour hand travels that much in one minute).
The minute hand accomplish 6 degree rotation in one minute
No to the problem at 7.20
Position of hour hand from 12(0 degree) is = 30*7 + 20*.5 = 210 + 10 = 220
Position of minute hand is 20*6 = 120 degrees from 12(0 degree)
The difference in angles is 100 degrees. Not divisible by 15.
Employing the same principle on the rest we get answer A.

Approach Solution 2:
Formula for angle between Hour ( H) and minute ( m ) hand is
(60H-11m)/2
Using this : 7.20 ->60*7 -11*20 /2 = 100 not divisible by 15
9:00 -> 180 divisible by 15
4:30 -> 105 divisible by 1

Approach Solution 3:
At 7:20, is the hour hand at 7 or a third between 7 and 8?
At 4:30, is the hour hand at 4 or mid way between 4 and 5?
We have to account for the little bit of distance covered by the hour hand too.
Employ Relative Speed here.
Minute hand covers 360 degrees in an hour.
Hour hand covers 360/12 = 30 degrees in an hour.
Speed of minute hand relative to hour hand is 360 - 30= 330 degrees per hour.
At 7 o'clock, the minute hand is 210 degrees behind the hour hand. In 20 minutes (at 7:20), it makes up 330/3 = 110 degrees. Now it will be 100 degrees behind the hour hand. The smaller angle between them is 100 degrees.
At 4 o'clock, the minute hand is 120 degrees behind the hour hand. In half an hour, it covers 330/2 = 165 degrees to get 45 degrees ahead of the hour hand. The smaller angle between them is 45 degrees.
At 9 o'clock, the angle between the two hands is 90 degrees.

“In a circular clock, the long hand is the radius of the”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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