## Clocks Questions - 1

Find the angle between the minute hand and hour hand of a clock when the time is 3:30 ?
• 70o
• 80o
• 75o
• 85o
• 105o
Explanation

At 3:30, Angle traced by the hour hand at 3:30 is, as each 1 hour = 30o.

3 $\displaystyle \frac{1}{2} \times 30^o = 105^o$

Angle traced by the minute hand, as each 1 minute = 6o.

30 minutes = 30 x 6o = 180o.

Angle between the hands = 180o â€“ 105o = 75o.

Workspace
Find the angle between the minute hand and hour hand of a clock when the time is 4:45 ?
• 142.5o
• 127.5o
• 270o
• 150o
• 145o
Explanation

At 4:45, Angle traced by the hour hand at 4:45 = $\displaystyle 4 \times 30^o = 120^o + \frac{3}{4}$ hr x 30o = 22.5o.

120 + 22.5 = 142.5o

Angle traced by the minute hand in 45 minutes = 45 x 6o = 270o

Angle between the hands = 270 â€“ 142.5 = 127.5o

Workspace
Find the reflex angle between the hands of a clock when the time is 3:45 ?
• 202.5o
• 157.5o
• 112.5o
• 270o
• 145o
Explanation

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 6o = 270o

Angle between the hands = 270 â€“ 112.5 = 157.5o

Reflex angle = 360 â€“ 157.5 = 202.5o

Workspace
At what time between 6 and 7 Oâ€™clock are the hands of a clock coincident ?
• 30 min past 6
• 31 min past 6
• $\displaystyle 31^{\frac{9}{11}}$ min past 6
• $\displaystyle 32^{\frac{8}{11}}$ min past 6
• $\displaystyle 32^{\frac{1}{2}}$ min past 6
Explanation

At 6 Oâ€™ 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.

Workspace
At what time between 6 and 7 Oâ€™clock will the hands of a clock be at right angle ?
• 15 min past 6 and 45 min past 6
• $\displaystyle 15^{\frac{5}{11}}$ min past 6 and $\displaystyle 45^{\frac{5}{11}}$ min past 6
• $\displaystyle 16^{\frac{2}{11}}$ min past 6 and $\displaystyle 49^{\frac{2}{11}}$ min past 6
• $\displaystyle 16^{\frac{1}{2}}$ min past 6 and $\displaystyle 49^{\frac{1}{2}}$ min past 6
• $\displaystyle 16^{\frac{4}{11}}$ min past 6 and $\displaystyle 49^{\frac{1}{11}}$ min past 6
Explanation

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 Oâ€™ Clock, hour hand is at 6 (on 30 min space), minute hand is at 12.

1st time: minute hand has to gain = 30 â€“ 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.

Workspace
At what time between 4 and 5 Oâ€™clock will the hands of a clock be in a straight line but not together. ? [opposite direction]
• 50 min past 4
• $\displaystyle 55^{\frac{5}{11}}$ min past 4
• 55 min past 4
• $\displaystyle 55^{\frac{4}{11}}$ min past 4
• $\displaystyle 54^{\frac{6}{11}}$ min past 4
Explanation

To be in straight line, the hands of a clock must me 30 minute spaces apart from each other.

At 4 Oâ€™ 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.

Workspace
At what time between 12 and 1 Oâ€™clock will the hands of a clock be in a straight line ?
• 12 :00 and 12:32 min
• $\displaystyle 12^{\frac{1}{2}}$ min and 12: $\displaystyle 32^{\frac{8}{11}}$
• 12:00 and 12 : $\displaystyle 32^{\frac{8}{11}}$ min
• 12 :00 and 12: 30 min
• 12:00 and 12 : $\displaystyle 31^{\frac{1}{2}}$ min
Explanation

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 Oâ€™ 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.

Workspace
At what time between 7 and 8 Oâ€™clock will the hands of a clock be in a straight line ?
• 7 : 05 min
• $\displaystyle 7 : 05^{\frac{5}{11}}$ min
• 7 : 10 min
• $\displaystyle 7 : 05^{\frac{1}{2}}$ min
• 7 : 15 min
Explanation

At 7 Oâ€™ 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.

Workspace
A clock is set right at 12:00 noon. The clock loses 15 min in 24 hours. What will be the true time when the clock indicates 6 pm, after 3 days ? [approx].
• 6 : 45 PM
• 6 H : 47 M : 30 sec AM
• 6 H : 47 M : 30 sec PM
• 6 H : 43 M : 20 sec PM
• 6 H : 49M : 12 sec PM
Explanation

Completed time on wrong clock at 6:00 pm after 3 days.

 From 12:00 am to 12:00 am Â = Â 24 x 3 days = Â 72 hours From 12:00 am to 6:00 pm Â = Â Â Â 6 hours Total Â = Â Â Â 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 Ã· 95 = 78.82105 (rounded off to 78.82).

Lost time = 78.82 â€“ 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).

Workspace
A clock is set right at 12 : 00 am. The clock loses 20 min in 24 hours. What will be the right time after 5 days, when this clock indicates 6 : 00 pm ? [approx]
• 7 H : 50 M : 45 sec PM
• 7 : 40 PM
• 7 H : 51 M : 24 sec PM
• 7 H : 22 M : 20 sec PM
• 7 H : 52 M : 30 sec PM
Explanation

Completed time on wrong clock at 6:00 pm after 5 days.

 12:00 am to 12:00 am Â = Â 24 Hours x 5 days = Â 120 hours 12:00 am to 6:00 pm Â = Â Â Â 12 hours Total Â = Â Â Â 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 Ã· 71 = 133.86.

Lost time = 133.86 â€“ 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.

Workspace

#### Practice Test Report

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## Clocks Fundamental Principles and Formulas

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 360o, 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 [360o Ã· 12] x 3 = 90o or 30o x 3 = 90o.

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 = 180o or 6o x 30 min = 180o.

### Important points:

• Hands of a clock coincide/together for 22 times in a day.
• Hands of a clock are on straight line for 44 times in a day. [Coincide 22 times + Opposite direction 22 times].
• Hands of a clock are at right angle for 44 times in a day [2 times in every hour]
• In a correct clock the two hands coincide at an interval of 65 $\displaystyle \frac{5}{11}$ min in every one hour.

Time: