Question

Consider a watch that gains uniformly, is $ 5 $ minutes slow on Sunday morning at $ 8 $ am, and the watch is $ 5 $ minutes and $ 48 $ seconds faster on the following Sunday night at $ 8 $ pm. When is the watch showing the correct value?
(A) Wednesday, it is $ 20 $ minutes past $ 7 $ pm
(B) Thursday, it is $ 15 $ minutes past $ 7 $ pm
(C) Wednesday, it is $ 29 $ minutes past $ 8 $ pm
(D) Thursday, $ 20 $ minutes past $ 7 $ pm

Answer 1

Hint: We must be clear about the number of hours in a day that is hours. We need to first find out the total number of hours from Sunday morning to following Sunday night, and calculate its total time gain. We must also utilize the information given about when the clock was slow, that will help us find out when exactly the clock showed the right time.

Complete answer:
Let us see what the total time from Sunday $ 8 $ am to the next Sunday $ 8 $ pm is;
From one Sunday to the next at $ 8 $ am it would complete $ 7 $ days, and additional $ 12 $ hours since the gain we are monitoring is till $ 8 $ pm the following Sunday.
 $ \Rightarrow 7\;days\;12\;hours $
 $ \Rightarrow 24 \times 7 + 12\;hours $
 $ \Rightarrow 168 + 12\;hours $
 $ \therefore $ Total time $ \Rightarrow 180\;hours $
They have mentioned to us that the watch is $ 5 $ minutes and $ 48 $ seconds faster on Sunday night, this means:
Watch gain $ = (5 + 5\dfrac{4}{5}) $ minutes $ = \dfrac{{54}}{5} $ minutes after $ 180 $ hours.
It is clear to us that $ \dfrac{{54}}{5} $ minute gain has happened in $ 180 $ hours.
 $ \therefore $ There is a $ 1 $ minute gain in $ 180 \times \dfrac{5}{{54}} $ hours.
So we can say that $ \Rightarrow $ $ 5 $ minutes will be gained in $ (5 \times 180 \times \dfrac{5}{{54}}) $ hours $ = 83 $ hours and $ 20 $ minutes
 $ \therefore $ Proper time will be reached after the clock has gained $ 5 $ minutes because it was 5 minutes slow initially, that is from Sunday $ 8 $ am after $ \Rightarrow 3\;days\;11\;hours\,\;20\;{\text{minutes}} $
To be exact we can say that the correct time is on Wednesday, $ 20 $ minutes past $ 7 $ pm.
$ \therefore $ The correct option will be option (A) Wednesday, it is $ 20 $ minutes past $ 7 $ pm.

Note:
If we need to find the angle between any two hands of a clock at a given time, we need to follow a certain procedure.
i. Make note of the number of hours, minutes or second at which the clock completes $ 1\;round = 360\;\deg $
ii. We must measure the angle formed by the slowest hand to the fastest moving hand in this order $ \to $ (hour $ < $ minute $ < $ second).
iii. Find the angle in degrees
iv. Subtract the degrees from the slower hand to the faster one to calculate the required degrees.