Solveeit Logo
Login

Question

Question: At what time, between nine o’clock and ten o’clock, are the hands of a clock in the opposite directi...

At what time, between nine o’clock and ten o’clock, are the hands of a clock in the opposite direction?
A. 99 hours 1781117\dfrac{8}{{11}} minutes
B. 99 hours 2191121\dfrac{9}{{11}} minutes
C. 99 hours 2391123\dfrac{9}{{11}} minutes
D. 99 hours 4891148\dfrac{9}{{11}} minutes
E. 99 hours 1641116\dfrac{4}{{11}} minutes

Explanation

Solution

We need to find the time at which the hands of a clock ( minute hand and hour hand ) are in the opposite direction not the whole day but between nine o’clock and ten o’clock . Between nine o’clock and ten o’clock ‘hour hand’ lies between 99 and 1010. In one hour ‘minute hand ’ gains 3300{330^0} or 5555 minute space .

Complete step by step answer:
We need to find at what time ‘minute hand’ will be at the opposite of ‘hour hand’ .
The large hand on a clock that points to the minute is called ‘minute hand’. The small hand denotes hour, which is why they are called ‘hour hand ’. At 99 o’clock ‘minute hand’ is at 12 and ‘hour hand ’ is at 9 . So at 99 o'clock the minute hand is behind the ‘hour hand ’ by 9×59 \times 5= 4545 minute space .

Minute hand will be the opposite of the hour hand if it is between 33 and 44 . As 33 and 44 is opposite of 99 and 1010. As ‘minute hand ’ between 33 and 44 means 1515 to 2020 minutes we can assume that ‘hour hand ’ will be near to 99.So ‘ minute hand’ has to gain 3×53 \times 5 = 1515 minute space as 33 is opposite to 99. In one hour ‘minute hand’ moves 3600{360^0}

In one hour ‘hour hand ’ moves 360012\dfrac{{{{360}^0}}}{{12}} = 300{30^0}. So In one hour ‘minute hand ’ gains 3600−300=3300{360^0} - {30^0} = {330^0} which means 5555 minutes space . Gain of 5555 minutes space achieved in 6060 minutes .So gain of 1515 minutes space achieved in 6055×15\dfrac{{60}}{{55}} \times 15 minutes = 18011\dfrac{{180}}{{11}} minutes = 1641116\dfrac{4}{{11}} minutes .

So at 99 hours 1641116\dfrac{4}{{11}} minutes both the two hands of the watch will be in opposite directions between nine o’clock and ten o’clock .

Therefore, option (E) is the correct answer.

Note: Careful about minute and minute space . minute space denotes the space or angle minute hand moves but minute represents time . Be attentive about calculation how much minute space ‘minute hand’ has to gain . Be extra careful about the portion where we deduce gain of 5555 minutes space achieved in 6060 minutes.