Clocks & Calendars: Aptitude Concepts

What is the question? A clock gains or loses time at a fixed rate. You are given the actual time and asked what the faulty clock will show.
Formula:

$\text{Total error}=\text{Error per hour}\times\text{Elapsed hours}$

Then:

$\text{Displayed time}=\text{Actual time}\pm\text{Error}$
Example: A clock gains 5 minutes per hour. It is correct at 7:00 AM. What will it show at 7:00 PM actual time?

Actual elapsed time:
```
7 AM → 7 PM = 12 hours
```
Gain:

$12\times5=60\text{ minutes}=1\text{ hour}$

Therefore:
```
Actual time = 7:00 PM
Gain        = 1 hour
```
$7:00\text{ PM}+1\text{ hour}=8:00\text{ PM}$

Answer: 8:00 PM

What is the question? A faulty clock shows a particular time. Find the actual time.
Formula:

$\text{Actual time}=\text{Displayed time}-\text{Gain}$

or

$\text{Actual time}=\text{Displayed time}+\text{Loss}$
Example: A clock gains 4 minutes per hour. It was correct at 8:00 AM. When it shows 2:00 PM, what is the actual time?

Clock shows:
```
8 AM → 2 PM = 6 hours
```
Gain:

$6\times4=24\text{ minutes}$

Therefore the actual time is:
```
2:00 PM - 24 minutes
= 1:36 PM
```
Answer: 1:36 PM

What is the question? Two clocks gain or lose time at different rates. Find when they will show the same time again.
Formula:
$=\text{Gain rate}+\text{Loss rate}$$ Then: $$\text{Time}=\frac{\text{Required difference}}{\text{Relative error rate}}$$$
Example: One clock gains 3 minutes per hour and another loses 2 minutes per hour. Both are correct at 12 noon. When will they next show the same time?

Every hour, their difference increases by:

$3+2=5\text{ minutes}$

For a 12-hour clock, they show the same reading again when their difference reaches:

$12\times60=720\text{ minutes}$

Therefore:

$\text{Time}=\frac{720}{5}=144\text{ hours}$

$144\text{ hours}=6\text{ days}$

Answer: 6 days later at 12 noon

What is the question? A clock gains or loses time over several days, and the question combines this with a date or day calculation.
Formula:

$\text{Total error}=\text{Error rate}\times\text{Elapsed time}$

For day shifts:

$\text{Odd days}=\text{Total days}\bmod7$
Example: A clock gains 2 minutes per hour. It is correct on Monday at 8:00 AM. How much will it have gained after 3 days?

3 days:

$3\times24=72\text{ hours}$

Total gain:

$72\times2=144\text{ minutes}$

$144\text{ minutes}=2\text{ hours }24\text{ minutes}$

Answer: The clock gains 2 hours 24 minutes.

The day calculation is separate:
```
3 days → 3 mod 7 = 3
Monday + 3 days = Thursday
```
Answer: Thursday at 8:00 AM actual time.

What is the question? First determine the actual time from a fast/slow clock, then find the angle between its hands.
Formula:

$\theta=|30H-5.5M|$
Example: A clock gains 2 minutes per hour. It was correct at 8:00 AM. When the clock shows 2:00 PM, find the angle between its hands.

Clock has run for 6 hours:

$\text{Gain}=6\times2=12\text{ minutes}$

Actual time:

$2:00-12\text{ min}=1:48\text{ PM}$

Now use:
```
H = 1
M = 48
```
$\theta=|30(1)-5.5(48)|$

$=|30-264|$

$=234^\circ$

Smaller angle:

$360-234=126^\circ$

Answer: $126^\circ$

What is the question? These questions combine two independent cycles, such as a faulty clock and a repeating calendar pattern, and ask when both conditions occur together.
Formula:

$\text{Required time}=\operatorname{LCM}(\text{relevant periods})$
Example: Two events repeat every 6 days and 8 days. After how many days will they occur together again?

Find the LCM:

$6=2\times3$

$8=2^3$

Therefore:

$LCM(6,8)=2^3\times3=24$

Answer: 24 days

This type is advanced and is much less common than standard clock-angle, fast/slow clock, and calendar questions.

Clocks and Calendars Concepts