1. Question:At what time between 2 and 3 o'clock will hands of a clock together ? 

    Answer
    At 2 o'clock, the hour hand is it 2 and the minute hand is at 12, i,e they are 10 min spaces apart.
    
    To be together the minute hand must gain 10 minutes over the hour hand.
    
    Now, 35 minutes are gained by it in 60 min.
    
    :. 10 minutes will be gained in` (60/55 xx 10)`
    
    :. The hands will coincide at `10 10/11` min past 2.






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  2. Question:At what time between 4 and 5 o'clock will the hands of a clock be at right angle ? 

    Answer
    At 4 o'clock the minute hand will be 20 min.spaces behind the hour hand.
    
    Now, when the two hands are at right angles, they are 15 min. spaces apart.
    
    So they are at right angles in following two cases.






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  3. Question:Find at what time between 8 and 9 o'clock will the hands of a clock be in the same straight line but not together. 

    Answer
    At 8 o'clock the hour hand is at 8 and the minute hand is at 12 i,e. the two hands are 20 min. spaces apart.
    
    To be in the same straight line but not together they will be 30 minute spaces apart.
    So, the minute hand will have to gain (30 - 20) = 10 minute spaces over the hour hand.
    
    55 minute spaces are gained in 60 min.
    
    10 minute spaces will be gained in` (60/55 xx 10)` min` = 10 10/11`
    
    :. The hands will be in the same straight line but not together at
    
    `10 10/11` min. part 8.






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  4. Question:At what time between 5 and 6 o'clock are the hands of a 3 minute apart ? 

    Answer
    At 5 o'clock, the minute hand is 25 min. space behind the hour hand.
    
    Case I: Minute hand is 3 min.spaces behind the hour hand.
    
    In this case, the minute hand has to gain (25 - 3) = 22 minute space 
    
    55 min are gained in 60 min.
    
    22 min. are gained in` (60/55 xx 22)` min. = 24 min
    
    :. The hands will be 3 min. apart at` 31 5/11` min. past 5.
    
    Case II : Minute hand is 3 min.spaces ahead of the hour hand.
    
    In this case, the minute hand has to gain (25 + 3) = 28 minute spaces.
    
    55 min.are gained in 60 min.
    
    28 min. are gained in` (60/55 xx 28)` `= 31 5/11` min.
    
    :. The hands will be 3 min. apart at `31 5/11` min. past 5.






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  5. Question:A watch which gains uniformly's is 5 min. slow at 8 o'clock in the morning on Sunday and it is 5 min. 48 sec. fast at 8 p.m. on following Sunday. When was it correct ? 

    Answer
    Time from 8 a.m. on Sunday to 8 p.m. on following Sunday
    
           = 7 days 12 hours = 180 hours.
    
    :. The watch gains `(5 + 5 4/5)` min or `54/5` min. in 180 hrs.
    
    Now` 54/5` min. are gained in 180 hrs.
    
    :. 5 min. are gained in `(180 xx 5/54 xx 5)` hrs.
    
    = 83 hrs 20 min. = 3 days 11 hrs 20 min.
    
    :. watch is correct 3 days 11 hrs 20 min. past 7 p.m.on Wednesday.
    
    :. It will be correct at 20 min.past 7 p.m. on Wednesday.






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