Question

A watch that gains 5 seconds in 3 minutes was set right at 7 a.m. In the afternoon of the same day, when the watch indicated quarter past 4 o'clock, the true time is:
(A) \[\text{59}\dfrac{\text{7}}{\text{12}}\text{min}\text{.}\,\text{past}\,\text{3}\]
(B) 4 p. m
(C) \[\text{58}\dfrac{\text{7}}{\text{11}}\text{min}\text{.}\,\text{past}\,\text{3}\]
(D) \[2\dfrac{3}{\text{11}}\text{min}\text{.}\,\text{past}\,\text{3}\]

Answer 1

Hint: It is given that the watch gains 5 seconds in minutes. It means when 3 minutes and 5 seconds of this clock is equal to 3 minutes of the correct clock. Here, we have our equation as 3 minutes and 5 seconds of this clock = 3 minutes of the correct clock. We also know that, \[60\sec =1\min \] . Use this and transform the equation in minutes into seconds as 185 seconds of this clock = 180 seconds of the correct clock. We know that, \[3600\sec =1hr\] . Use this and transform the equation in seconds into hours as \[\dfrac{37}{720}\] hr of this clock = \[\dfrac{1}{20}\] hr of the correct clock. Use the unitary method and get the time that is to be covered in the correct clock if this clock passes 1 hour. The clock was set at 7 a.m and when the watch indicated quarter past 4 o'clock then get the time that must have been passed in this watch. The difference in time from 7 a.m to 4:15 p.m is \[\dfrac{37}{4}\] hr. Now, use the equation 1 hr of this clock = \[\left( \dfrac{1}{20}\times \dfrac{720}{37} \right)\] hr of the correct clock, get the time that is to be covered in the correct clock if this clock passes \[\dfrac{37}{4}\] hour. Then add that hour of the correct clock at 7 a.m and get the time in the correct clock.

Complete step-by-step answer:
According to the question, it is given that we have a watch that gains 5 seconds in minutes. It means when 3 minutes and 5 seconds of this clock is equal to 3 minutes of the correct clock.
3 minutes and 5 seconds of this clock = 3 minutes of the correct clock ………………………..(1)
We know that, \[3600\sec =1hr\] …………………..(2)
Now, simplifying equation (2), we get
\[1\sec =\dfrac{1}{3600}hr\] …………………………..(3)
We also know that, \[60\sec =1\min \] …………………..(4)
Now, using equation (4) and transforming equation (1), we get
3 minutes and 5 seconds of this clock = 3 minutes of the correct clock
\[\left( 3\times 60+5 \right)\] seconds of this clock = \[3\times 60\] seconds of the correct clock
185 seconds of this clock = 180 seconds of the correct clock ………………….(5)
Now, using equation (3) and transforming equation (5), we get
\[\left( 185\times \dfrac{1}{3600} \right)\] hr of this clock = \[\left( 180\times \dfrac{1}{3600} \right)\] hr of the correct clock
\[\dfrac{37}{720}\] hr of this clock = \[\dfrac{1}{20}\] hr of the correct clock
1 hr of this clock = \[\left( \dfrac{1}{20}\times \dfrac{720}{37} \right)\] hr of the correct clock …………………………(6)
The difference of time from 7 a.m to 4:15 p.m = 9 hrs 15 min …………………..(7)
We know that, \[60\min =1hr\] ……………………….(8)
Now, simplifying equation (8), we get
\[1\min =\dfrac{1}{60}hr\] …………………..(9)
Using equation (9) and transforming equation (7), we get
The difference of time from 7 a.m to 4:15 p.m = 9 hr 15 min = 9 hr + \[\dfrac{15}{60}\] hr = 9 hr + \[\dfrac{1}{4}\] hr = \[\dfrac{37}{4}\] hr ………………………………(10)
Using equation (6), we can get the number of hours in the correct clock with respect to \[\dfrac{37}{4}\] hr of this clock.
\[\dfrac{37}{4}\] hr of this clock = \[\left( \dfrac{1}{20}\times \dfrac{720}{37}\times \dfrac{37}{4} \right)\] hr of the correct clock
\[\dfrac{37}{4}\] hr of this clock = \[\left( \dfrac{720}{80} \right)\] hr of the correct clock
\[\dfrac{37}{4}\] hr of this clock = 9 hr of the correct clock ………………………..(11)
The correct clock would pass 9 hr if \[\dfrac{37}{4}\] hr passed in this clock.
The clock was set at 7 a.m and after 9 hr the correct clock would show the time at 4 p.m.
Hence, the correct option is (B).

Note: This question involves multiple conversions of units of time. So, there is a chance of calculation mistakes while the conversion of time in one unit to other units. To rectify this, we have to be sincere while doing calculations for the conversion of time in one unit to other units.