Clock - Aptitude MCQ Questions and Solutions with Explanations

Angle traced by the minute hand in 5 minutes.
$$\eqalign{ & = {\left( {\frac{{360}}{{60}} \times 5} \right)^ \circ } \cr & = {30^ \circ }{\text{ }} \cr} $$

At 3 o'clock, the minute hand is 15 minute spaces behind the hour hand.
Thus, the minute hand has to gain (15 + 18) = 33 minute spaces
55 minutes are gained in 60 minutes.
33 minutes are gained in $$\left( {\frac{{60}}{{55}} \times 33} \right)$$   = 36 minutes
∴ The hands will be 18 minutes spaces apart at 3 : 36 pm.

Angle traced by the minute hand in 2 hours 20 minutes,i.e.,
$$\eqalign{ & = 140{\text{ minutes}} \cr & = {\left( {\frac{{360}}{{60}} \times 140} \right)^ \circ } \cr & = {840^ \circ } \cr} $$

Angle swept by the minute hand in 35 minutes.
$$\eqalign{ & = {\left( {\frac{{360}}{{60}} \times 35} \right)^ \circ } \cr & = {210^ \circ } \cr} $$
∴ Required area = Area of a sector of a circle with radius 10 cm and central angle 210°
$$\eqalign{ & = \frac{{\pi {r^2}\theta }}{{360}} \cr & = \left( {\frac{{22}}{7} \times 10 \times 10 \times \frac{{210}}{{360}}} \right){\text{c}}{{\text{m}}^2} \cr & = \frac{{550}}{3}{\text{c}}{{\text{m}}^2} \cr & = 183\frac{1}{3}{\text{c}}{{\text{m}}^2} \cr} $$

Let angle between the hands of clock be x
When the time is 4 : 25 am
Where [M = minutes and H = hours]
Required angle
$$\eqalign{ & = {\text{30}}\left( {\frac{{\text{M}}}{5} - {\text{H}}} \right) - \frac{{\text{M}}}{2} \cr & = 30\left( {\frac{{25}}{5} - 4} \right) - \frac{{25}}{2} \cr & = 30\left( {\frac{{25 - 20}}{5}} \right) - \frac{{25}}{2} \cr & = 30\left( {\frac{5}{5}} \right) - \frac{{25}}{2} \cr & = 30 - \frac{{25}}{2} \cr & = \frac{{60 - 25}}{2} \cr & = \frac{{35}}{2} \cr & = 17{\frac{1}{2}^{° }} \cr} $$

The watch loses $$\frac{1}{2}$$ hour each hour. So, it must have take 8 hours to show 4 pm from 12 noon.
Thus, it stopped at 8 pm.
So, the correct time is 5 hours ahead of 8 pm, i,e., 1 am.

Number of rotation
$$ = \frac{{72}}{{12}} = 6$$

In 1 hour, the hour hand make the angle of 30°
The hour hand make the angle of x in $${\text{8}}\frac{{30}}{{60}}$$ hours
$$\eqalign{ & \Rightarrow x = {\left( {30 \times 8\frac{1}{2}} \right)^ \circ } \cr & \Rightarrow x = {\left( {30 \times \frac{{17}}{2}} \right)^ \circ } \cr & \Rightarrow x = {255^ \circ } \cr} $$
The minute hand make the angle in 1 minute = 6°
Minute hand makes the angle in 30 minutes
$$\eqalign{ & {\text{ = }}{\left( {6 \times 30} \right)^ \circ } \cr & = {180^ \circ } \cr & {\text{Required angle}} \cr & = {255^0} - {180^0} \cr & = {75^0} \cr} $$

Time from 12 noon to 9 : 38 A.M.
= 12 hours + 9 hours 38 minutes
= 21 hours 38 minutes
$$\eqalign{ & {\text{ = 21}}\frac{{38}}{{60}}{\text{ hours}} \cr & {\text{ = 21}}\frac{{19}}{{30}}{\text{ hours}} \cr & {\text{ = }}\frac{{649}}{{30}}{\text{hours}} \cr} $$
Angle traced by the hour hand in 12 hours = 360°
Angle traced by the minute hand in
$$\eqalign{ & \Leftrightarrow \frac{{649}}{{30}}{\text{hours}} \cr & = {\left( {\frac{{360}}{{12}} \times \frac{{649}}{{30}}} \right)^ \circ } \cr & = {649^ \circ } \cr} $$

Number of rounds completed by the minute hand in 3 days 5 hours
$$\eqalign{ & = \left( {3 \times 24 + 5} \right) \cr & = 77 \cr} $$
Number of rounds completed by the hour hand in 3 days 5 hours
$$\eqalign{ & = \left( {3 \times 2 + \frac{5}{{12}}} \right) \cr & = 6\frac{5}{{12}} \cr} $$
∴ Difference between the distance traversed
$${\text{ = }}\left[ {77 \times \left( {2 \times \frac{{22}}{7} \times 10} \right) - 6\frac{5}{{12}} \times \left( {2 \times \frac{{22}}{7} \times 7} \right)} \right]{\text{cm}}$$
$$\eqalign{ & = \left( {4840 - 282.33} \right){\text{ cm}} \cr & = 4557.67{\text{ cm}} \cr} $$