Clock - Aptitude MCQ Questions and Solutions with Explanations

In 12 hours, they are at right angles 22 times.
∴ In 24 hours, they are at right angles 44 times.

$$\eqalign{ & {\text{Angle}}\,{\text{traced}}\,{\text{by}}\,{\text{hour}}\,{\text{hand}}\,{\text{in}}\,\frac{{17}}{2}\,{\text{hrs}} \cr & {\text{ = }}\,{\left( {\frac{{360}}{{12}} \times \frac{{17}}{2}} \right)^ \circ } \cr & = 255^ \circ \cr & {\text{Angle}}\,{\text{traced}}\,{\text{by}}\,{\text{min}}{\text{.}}\,{\text{hand}}\,{\text{in}}\,{\text{30}}\,{\text{min}} \cr & = {\left( {\frac{{360}}{{60}} \times 30} \right)^ \circ } \cr & = 180^ \circ \cr & \therefore {\text{Required}}\,{\text{angle}} \cr & = {\left( {255 - 180} \right)^ \circ } = {75^ \circ } \cr} $$

The hands of a clock point in opposite directions (in the same straight line) 11 times in every 12 hours. (Because between 5 and 7 they point in opposite directions at 6 o'clcok only).
So, in a day, the hands point in the opposite directions 22 times.

At 4 o'clock, the hands of the watch are 20 min. spaces apart.
To be in opposite directions, they must be 30 min. spaces apart.
∴ Minute hand will have to gain 50 min. spaces.
55 min. spaces are gained in 60 min.
50 min. spaces are gained in
$$\eqalign{ & \left( {\frac{{60}}{{55}} \times 50} \right){\text{min}}{\text{.}}\,{\text{or}}\,54\frac{6}{{11}}\,{\text{min}}. \cr & \therefore {\text{Required}}\,{\text{time}} \cr & = 54\frac{6}{{11}}{\text{min}}{\text{.}}\,{\text{past}}\,4 \cr} $$

To be together between 9 and 10 o'clock, the minute hand has to gain 45 min. spaces.
55 min. spaces gained in 60 min.
45 min. spaces are gained in
$$\left( {\frac{{60}}{{55}} \times 45} \right){\text{min}}{\text{.}}{\kern 1pt} \,{\text{or}}{\kern 1pt} \,49\frac{1}{{11}}{\kern 1pt} {\text{min}}.$$
∴ The hands are together at $$49\frac{1}{{11}}$$ min. past 9

At 3 o'clock, the minute hand is 15 min. spaces apart from the hour hand.
To be coincident, it must gain 15 min. spaces.
55 min. are gained in 60 min.
15 min. are gained in $$\left( {\frac{{60}}{{55}} \times 15} \right)$$   min.
= $$16\frac{4}{{11}}$$ min.
∴ The hands are coincident at $$16\frac{4}{{11}}$$ min. past 3

The hands of a clock coincide 11 times in every 12 hours (Since between 11 and 1, they coincide only once, i.e., at 12 o'clock).
AM
12:00
1:05
2:11
3:16
4:22
5:27
6:33
7:38
8:44
9:49
10:55

PM
12:00
1:05
2:11
3:16
4:22
5:27
6:33
7:38
8:44
9:49
10:55
The hands overlap about every 65 minutes, not every 60 minutes.
∴ The hands coincide 22 times in a day.

In 12 hours, the hands coincide or are in opposite direction 22 times.
∴ In 24 hours, the hands coincide or are in opposite direction 44 times a day.

Time from 12 pm on Monday to 2 pm on the following Monday = 7 days 2 hours
= 170 hours
∴ The watch gains \[\left( {2 + 4\frac{4}{5}} \right)\]   minutes or \[\frac{{34}}{5}\]   minutes in 170 hours.
Now, $$\frac{{34}}{5}$$ minutes are gained in 170 hours
∴ 2 minutes are gained in \[\left( {170 \times \frac{5}{{34}} \times 2} \right)\]   hours = 50 hours
∴ Watch is correct 2 days 2 hours after 12 pm on Monday i.e., it will be correct at 2 pm on Wednesday.

Number of minutes in a day
= (24 × 60) = 1440
∴ Required percentage
$$\eqalign{ & = \left( {\frac{5}{{1440}} \times 100} \right)\% \cr & = \frac{{50}}{{144}}\% \cr} $$