Clock - Aptitude MCQ Questions and Solutions with Explanations

One clock show 10 pm, on 21^st January 2010
One clock gains = 2 minutes
Other clock loses = 5 minutes
Time period between 10 pm and 4 pm = 18 hours
∴ Required difference
= (2 × 18 + 5 × 18 ) minutes
= 126 minutes

Time gained in 1 hour = 6 minutes
Time gained in 6 hours = (6 × 6) minutes = 36 minutes
After 6 hours, the correct time is 11 : 00 am and the watch will show 11 : 36 am.

Time from 1 pm on Wednesday to 1 pm on Thursday = 48 hours
So, the watch gains (1 + 2) minute or 3 minutes in 48 hours.
Now, 3 minutes are gained in 48 hours
So, 1 minute is gained in $$\left( {\frac{{48}}{3}} \right)$$   = 16 hours.
Thus, the watch showed the correct time 16 hours after 1 pm on Tuesday, i.e., 5 am on Wednesday

To be together between 8 am and 9 am, the minute hand has to gain 40 minutes spaces.
55 minutes spaces are gained in 60 minutes.
40 minutes space are gained in $$\left( {\frac{{60}}{{55}} \times 40} \right)$$  minutes = $${\text{43}}\frac{7}{{11}}$$  minutes
So, Henry started his trip at $${\text{43}}\frac{7}{{11}}$$  minutes past 8 am.
Now, to be 180° apart, the hands must be 30 minutes spaces apart.
At 2 pm, they are 10 minutes spaces apart.
∴ The minute hand will have to gain (10 + 30) = 40 minutes spaces.
As calculate above, 40 minutes spaces are gained in $${\text{43}}\frac{7}{{11}}$$  minutes.
So, Henry's trip ended at $${\text{43}}\frac{7}{{11}}$$  minutes past 2 pm
∴ Duration of travel = Duration from $${\text{43}}\frac{7}{{11}}$$  minutes past 8 am to $${\text{43}}\frac{7}{{11}}$$  minutes past 2 pm = 6 hours

Since the time read by the lady was 57 minutes earlier than the correct time, so the minute hand is (60 - 57) = 3 minutes spaces behind the hour hand.
Now, at 5 o'clock, the minute hand is 25 minutes spaces behind the hour hand.
To be 3 minutes spaces behind, it must gain (25 - 3) = 22 minutes spaces.
55 minutes spaces are gained in 60 minutes.
22 minutes spaces are gained in $$\left( {\frac{{60}}{{55}} \times 22} \right)$$   = 24 minutes
Hence, the correct time was 24 minutes past 5.

Explanation in Detail:

To determine how many times the hour hand and minute hand of a clock form a right angle between 1:00 pm and 10:00 pm, we analyze their angular positions over this period.

1. Calculating Angular Movements:
- Minute Hand: The minute hand moves 360 degrees in 60 minutes, so in 9 hours (from 1:00 pm to 10:00 pm), it covers:
     \[ 360 \text{ degrees/hour} \times 9 \text{ hours} = 3240 \text{ degrees} \]
- Hour Hand: The hour hand moves 30 degrees in 60 minutes (or 0.5 degrees per minute), covering:
    \[ 30 \text{ degrees/hour} \times 9 \text{ hours} = 270 \text{ degrees} \]
2. Relative Angular Distance:
- The difference in their angular positions over 9 hours is:
    \[ 3240 \text{ degrees (minute hand)} - 270 \text{ degrees (hour hand)} = 2970 \text{ degrees} \]
3. Calculating Right Angles:
- A right angle is formed every 180 degrees.
- Therefore, the number of times they form a right angle is:
     \[ \frac{2970 \text{ degrees}}{180 \text{ degrees}} = 16.5 \]
Rounding down, they form a right angle 16 times.
4. Identifying Times of Right Angles:
- The first right angle occurs at 1:21.8181... pm.
- Subsequent right angles occur approximately every 32.7878... minutes until the last right angle at 9:32.7272... pm.
5. List of Times:
- The hour and minute hands form a right angle at the following times:
- 1:21.8181 pm
- 1:54.5454 pm
- 2:27.2727 pm
- 3:00 pm
- 3:32.7272 pm
- 4:05.4545 pm
- 4:38.1818 pm
- 5:10.9090 pm
- 5:43.6363 pm
- 6:16.3636 pm
- 6:49.0909 pm
- 7:21.8181 pm
- 7:54.5454 pm
- 8:27.2727 pm
- 9:00 pm
- 9:32.7272 pm
Therefore, the hour hand and minute hand of the clock form a right angle 16 times between 1:00 pm and 10:00 pm, as calculated based on their angular movements and the criteria for forming right angles.

After 12 days, i.e., after 12 × 24 hours clock A will gain 48 minutes and will show 12 : 48 noon.
After 12 days, i.e., after 12 × 24 hours clock B will loose 16 minutes and will show 11 : 44 am
The two clocks will show the same time after time after 135 days.
The time difference has to be 12 hours between then = 720 minutes.
A will gain 540 minutes in 135 days.
B will loose 180 minutes in 135 days, total 720 minutes.
Further if we consider only time then the problem becomes simpler
Total difference of minutes between the times shown by the clocks after 36 hours
⇒ $$\frac{{16}}{3}$$   minutes difference in 1 day
⇒ 12 × 60 minutes difference in $$\frac{3}{{16}}$$ × 12 × 60 = 135 days
∴ 12 noon, after 135 days

Time interval from 9 am on Monday to 8 : 30 pm on Wednesday.
$$\eqalign{ & {\text{ = }}\left( {24 \times 2.5} \right) - {\text{0:30 hours }} \cr & {\text{ = 60}} - {\text{0}}{\text{:30 hours}} \cr & {\text{ = 59 hours 30 minutes}} \cr & = 59\frac{{30}}{{60}} \cr & = 59\frac{1}{2} \cr & = \frac{{119}}{2}{\text{ hours}} \cr & {\text{Also 24 hours}} - {\text{12 minutes}} \cr & = {\text{23 hours 48 minutes}} \cr & = 23 + \frac{{48}}{{60}} \cr & = 23\frac{4}{5} \cr & = \frac{{119}}{5}{\text{ hours}} \cr & \therefore \frac{{119}}{2}{\text{ hours of this clock}} \cr & = \frac{{24 \times 5}}{{119}} \times \frac{{119}}{2} \cr & = 60{\text{ hours}} \cr & \left( {60 - \frac{{119}}{2}} \right){\text{ hours}} \cr & {\text{ = }}\frac{{120 - 119}}{2}{\text{ hours}} \cr & {\text{ = }}\frac{1}{2}{\text{ hours}} \cr & = 30{\text{ minutes}} \cr} $$
Hence, the correct time is 30 minutes after 8:30 pm i.e., 9 pm

There are 8 intervals in 9 tinging 10 intervals in 11 tinging.
Time duration of 8 intervals = 9 seconds
∴ Required time = Duration of 10 intervals
$$\eqalign{ & = \left( {\frac{9}{8} \times 10} \right){\text{ seconds}} \cr & = 11.25{\text{ seconds}} \cr} $$

$$\eqalign{ & {\text{Time lost in }}6\frac{1}{2}{\text{ hours}} \cr & = {\text{ }}\left( {6\frac{1}{2} \times 4} \right)\sec \cr & = 26\sec \cr} $$
Correct time after $${\text{6}}\frac{1}{2}$$ hours
= 7 hours 40 minutes 6 seconds + 6 hours 30 minutes
= 14 hours 10 minutes 6 seconds
Time show by the clock
= 14 hours 10 minutes 6 seconds - 26 sec
= 14 hours 9 minutes 40 seconds