At 3:30, Angle traced by the hour hand at 3:30 is, as each 1 hour = 30^{o}.
3 $\displaystyle \frac{1}{2} \times 30^o = 105^o$
Angle traced by the minute hand, as each 1 minute = 6^{o}.
30 minutes = 30 x 6^{o} = 180^{o}.
Angle between the hands = 180^{o} – 105^{o} = 75^{o}.
At 4:45, Angle traced by the hour hand at 4:45 = $\displaystyle 4 \times 30^o = 120^o + \frac{3}{4}$ hr x 30^{o} = 22.5^{o}.
120 + 22.5 = 142.5^{o}
Angle traced by the minute hand in 45 minutes = 45 x 6^{o} = 270^{o}
Angle between the hands = 270 – 142.5 = 127.5^{o}
At 3:45, Angle traced by the hour hand at 3:45 is = $\displaystyle 3 \times 30^o = 90^o + \frac{3}{4}$ hr $\displaystyle \times\ 30^o = 22.5^o = 112.5^o$
Angle traced by the minute hand in 45 minutes = 45 x 6^{o} = 270^{o}
Angle between the hands = 270 – 112.5 = 157.5^{o}
Reflex angle = 360 – 157.5 = 202.5^{o}
At 6 O’ Clock hour hand is at 6, minute hand is at 12.
To be coincident minute hand has to gain 30 minutes.
As minute hand gains 55 minutes over hour hand in 60 minutes, then 30 minutes in
$\displaystyle \frac{60}{55} \times 30 => \frac{12}{11} \times 30 = \frac{360}{11} = 32^{\frac{8}{11}}$ min past 6.
The hands of a clock, be at right angle for two times in every hour,
To be in right angle, the hands of a clock must be 15 minute spaces apart,
At 6 O’ Clock, hour hand is at 6 (on 30 min space), minute hand is at 12.
1st time: minute hand has to gain = 30 – 15 = 15 minute spaces.
$\displaystyle \frac{60}{55} \times 15 = \frac{12}{11} \times 15 = 16^{\frac{4}{11}}$ = minutes past 6
2nd time: minute hand has to gain => 30 + 15 = 45 minute spaces.
45 minnutes are gained in $\displaystyle \frac{60}{55} \times 45 => \frac{12}{11} \times 45 = 49^{\frac{1}{11}}$ = minutes past 6.
To be in straight line, the hands of a clock must me 30 minute spaces apart from each other.
At 4 O’ Clock hour hand is at 4 (on 20 min space), minute hand is at 12.
To be in opposite direction the minute hand has to gain 20 + 30 = 50 minute spaces.
50 minutes are gained in => $\displaystyle \frac{60}{55} \times 50 = \frac{12}{11} \times 50 = 54^{\frac{6}{11}}$ = minutes past 4.
In every hour, both the hands of a clock are on straight line for two times, in an hour
1st time when they are coincident, 2nd time when they are in opposite direction
Apparently at 12:00 O’ Clock both hour hand and minutes hand are coincident/straight-line.
2nd time [opposite direction] = to be on straight line minute hand has to gain 30 min spaces
Then, $\displaystyle \frac{60}{55} \times 30 = \frac{12}{11} \times 30 = 32^{\frac{8}{11}}$ minutes past 6.
At 7 O’ Clock minute hand will be 25 minute spaces ahead of hour hand,
To be on straight line [opposite direction] the minute hand has to gain 5 minutes.
Then, $\displaystyle \frac{60}{55} \times 5 = \frac{12}{11} \times 5 = 5^{\frac{5}{11}}$ minutes past 7.
Completed time on wrong clock at 6:00 pm after 3 days.
From 12:00 am to 12:00 am | = | 24 x 3 days = | 72 hours | |||
From 12:00 am to 6:00 pm | = | 6 hours | ||||
Total | = | 78 hours |
Wrong clock : Right clock.
23 : 45 hours = 24 hours.
Or
$\displaystyle \frac{95}{4}$ hours = 24 hours.
Time gained by the correct clock = 78 x 24 x $\displaystyle \frac{4}{95}$ = 7488 ÷ 95 = 78.82105 (rounded off to 78.82).
Lost time = 78.82 – 78 = 0.82 hours => 0.82 x 60 = 49.2 hours or 49 minutes 12 sec (approx).
Correct time = 6:00 pm + 49 minutes 12 sec = 6 H : 49 M : 12 sec PM. (Nearest time).
Completed time on wrong clock at 6:00 pm after 5 days.
12:00 am to 12:00 am | = | 24 Hours x 5 days = | 120 hours | |||
12:00 am to 6:00 pm | = | 12 hours | ||||
Total | = | 132 hours |
Wrong clock : Right clock.
23 : 40 hours = 24 hours.
Or
$\displaystyle \frac{71}{3}$ hours = 24 hours.
Time gained by the right clock = 132 x $\displaystyle \frac{3}{71}$ x 24 = 9504 ÷ 71 = 133.86.
Lost time = 133.86 – 132 = 1.86 hours = 1.86 x 60 = 1 Hour 51 min 24 seconds.
Time = 6:00 pm + 1 hr 51 m 24 sec = 7 H : 51 M : 24 sec PM.
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1. The face a Clock/watch look like a circle in shape circumference is divided into 60 equal parts called minute spaces.
2. A clock has two hands the smaller one is the hour hand, while the larger one is the minute hand.
3. In 60 minutes, the minute hand gains 55 minutes over hour hand.
4. In every hour both the hands coincide once.
5. The hands are in the straight line when they coincident or in opposite direction.
6. To be in right angle the hands of a clock must be 15 minute spaces apart.
7. To be in opposite direction the hands of a clock must be 30 minute spaces apart from each other.
8. As a clock in circle in shape, it is divided into 360^{o}, Each hour = $\displaystyle \frac{360^o}{12} = 30^o$.
For Ex. If the time is 3 O clock, the hour hand will be on 3 or points towards 3 Minute hand will be on 12.
Now the angle traced by the hour hand is [360^{o} ÷ 12] x 3 = 90^{o} or 30^{o} x 3 = 90^{o}.
For each minute = $\displaystyle \frac{360^o}{60} = 6^o$.
9. Angle traced by the minute hand:
Ex: At 12:30 the hour hand will be on 12, minute hand will be on 6, then angle traced by the minute hand is = $\displaystyle \frac{360^o}{60}$ = 6 = 180^{o} or 6^{o} x 30 min = 180^{o}.
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