Yolanda is proud of her new watch, as its minute and hour

Question

Yolanda is proud of her new watch, as its minute and hour hands are in continuous movement. At what time between 8 and 9 o'clock do the two hands come closest to forming a 60 degree angle?

(A) 8:31:50 (B) 8:32:45 (C) 8:54:10 (D) 8:54:12 (D) 8:54:58

ps_dahiya · Answer

Let minute hand is showing x minutes and seconds hand is showing y seconds.

Taking 12 at the reference point 
Angle made by hour hand = H = (240 + x/2 + y/120)
Angle made by mnute hand = M = (6x + y/10) 

H - M = 240 + x/2 -6x + y/120 - y/10
= 240 - (660x-11y)/120

Now H-M must be closest to 60. Put values of x and y from answers and see which one is closest to 60.

Long calculations......... Stuck....
Is there a shorter method?

kevincan · Answer

Forget about seconds- you only need one variable!

ps_dahiya · Answer

OK I got you.

Taking 12 at the reference point
Angle made by hour hand = H = 240 + x/2
Angle made by mnute hand = M = 6x

H - M = 240 +x/2 - 6x
= 240-11x/2
|H-M| = 60
so 
1. 240-11x/2 = 60
x = 360/11 = 32.73 minutes i.e 8:32:44

2. -240 + 11x/2 = 60
x = 600/11 = 54.54 minutes i.e around 8:54:32 seconds

So closest to these two times is B.

Zooroopa · Answer

Yes, the explanation is all right.

Take the required time as T minutes.

Find out the 'per-minute' figures for the long hand and the short hand:

Long Hand: (360-0)/60 = 6 degrees/min
Short Hand: (270-240)/60 = 0.5 degress/min (The position at 8 o' clock is at 240 degrees)

Now formulate your equation:

The difference is nearly 60 degrees.

So use the absolute value in your equation.

|6T- (240 + 0.5T)| = 60

Solve for the conditions 5.5T >= or < 240

For 5.5T >= 240 ---> T = 54.54 minutes or 54 mins 32.72 secs.

For 5.5T < 240 -----> T = 32.72 mins or 32 mins 43.63 secs (this is off B by a little over 1 sec)

lan583 · Answer

dahiya where did you get x/2 in your equation?

heman · Answer

Hr hand moves by 30deg for each hr
min hand moves by 6deg for each min
A) 8 'o' clock = 240 deg
 31minutes = 186 deg
 diff = 54deg
B) similarly difference = 58deg
C) diff = 84deg
D) close to C
E) close to C

Hence B

Heman

ps_dahiya · Answer

Angle made by hour hand with the line from center to 12 number.

1. For 8 hours, angle = 8 * 30 = 240
2. For x minutes, angle = 30x/60 = x/2. Every 60 minutes hour hand gains 30 degrees. So in x minutes hour hand will gain 30x/60.

nookway · Answer

This is how I approached it with some help from a friend.
 
Mark degrees around the dial. Let 12 oâ€™clock be 0 degrees, 1 oâ€™clock be 30 degrees, 2 oâ€™clock be 60 degrees, etc. The minute hand moves 360 degrees per hour, or 6 degrees per minute. The hour hand moves 360 degrees in 12 hours, or 30 degrees an hour, or 1/2 degree per minute.

Say the angle is negative if the minutehand is behind the hour hand, positive if it is ahead of the hour hand. At 8:00 letâ€™s say the angle is -240 degrees. The angle increases (toward +infinity) 5.5 degrees per minute. So the angle at m minutes after 8:00 is -240 + 5.5m degrees.

When is the angle -60 degrees?

-60 = -240 + 5.5m

180 = 5.5m

m = 180/5.5

m = 32.727272 â€¦ = 32 minutes 43.636363 â€¦ seconds

 
When is the angle +60 degrees?

60 = -240 + 5.5m

300 = 5.5m

m = 300/5.5

m = 54.545454 â€¦ = 54 minutes 32.727272 â€¦ seconds

 

Of the choices, 8:32:45 is closest to one of these. So (B) is the answer.

ivymba · Answer

B it is , but a simpler solution

1 unit of a watch is 360/60 = 6 degrees
for 60 degrees you need 10 units i.e 10 * 6

Just look at the values when the unit difference is is closest to 10 
And B has the closest one.
A) 50 - 31 = 29 units
B) 45 -32 = 13 units
C) 6 + 10 = 16 units
D) 6 + 12 = 18 units
E) 58 - 54 = 4 units

ivymba · Answer

I am sorry I misread the questions as always . Instead of hour and minutes I was doing Munutes hand and seconds hand. 
No wonder it didn't make sense. 

1 unit of a watch is 360/60 = 6 degrees 
for 60 degrees you need 10 units i.e 10 * 6 

hours hand is approximate angle relative to 12 is = 40 * 6 = 240 

(A) 8:31:50 minutes hand close to 31 * 6 = 186 
(B) 8:32:45 minutes hand close to 32 * 6 = 192 B is closest to 240 
(C) 8:54:10 minutes hand close to 54 * 6 = 324 
(D) 8:54:12 minutes hand close to 54* 6 = 324 
(D) 8:54:58 minutes hand close to 54 * 6 = 324

Rayn · Answer

Here's my solution.

In 1hr (60min), the minute hand moves 360 degrees and the hour hand moves 30 degrees.

The solution to the problem, therefore, would be the the distance in the degrees that the minute hand has not covered plus the distance that the hour hand has covered.

Beginning at 8:00, the minute hand has 240 degrees to cover. The hour hand has not moved.

At 8:31:50, the minute hand has moved almost 32min or 192 degrees (32/60 * 360 = 192) leaving a distance of 48 degrees uncovered (240 - 192). In the same time, the hour hand has moved 16degrees (32/60 * 30)
48 + 16 = 64 degrees.

At 8:32:45, the minute hand has moved 33min which, using the same reasoning above, is equal to 44 degrees uncovered by the minute hand  . The hour hand has moved about 17 degrees. (44 + 17 = 61 degrees)  My Answer 

At 8:54:10, closer to 9, the calc. is reversed.
The minute hand has moved 324 degrees (54/60 * 360). 324 - 240 = 84 degrees which is the distance covered by the min. hand past the hour hand. The hour hand has moved 27 degrees (54/60 * 30). 84 - 27 = 57 degrees, the distance between the hands.

Performing the same calcs for the other options.
D is close to 57
E is close to 63

Quite deep reasoning, I might be wrong...may have guessed A on the test. Any easier approaches.

old_dream_1976 · Answer

how about this 

Reject C, D and E 8:54 would make an angle more than 90-6 = 84 

Reject A because by the time the min hand reaches 6 position the hour hand has moved 15 degrees making the angle between hands > 60 +15-12 = 63 degrees 

B is 55 sec of more catch up better than A 60 +15 -17.xxxx=62.xxxxxx

so b

Yolanda is proud of her new watch, as its minute and hour

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