A boy started reading sometime between 3PM and 4PM and stopped reading sometime between 5PM and 6PM. He found that the hands of the watch interchanged their places. When did the boy started reading? When did he stopped reading? How long did he read? [Hint : When an hour's needle moves one hour, minute's needed moves 60 minutes.
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1 Answer

The start time is 3:M where M is the minutes. The finish time is therefore M/5:15 approx where 5<M/5<6, so 25<M<30. That gives us 3:25 as the approx start time and 5:15 as the approx finish time.

However, 25 minutes is 25/60=5/12 of an hour and the hour hand will therefore move 5/12 of the distance between 3 and 4 on the clock face, which is divided into 5 minute divisions. 5/12*5=25/12=2 1/12 minutes. This changes the finish time to 5hr (15+2 1/12)mins=5:17 1/12 or 5:205/12. But, this takes us back to the start time, because 205/12 minutes into the hour means that the hour hand moves 205/(12*60)*5=205/144 minutes after 25 past 3. That revises the start time to 25+205/144=3805/144=26.42 minutes approximately.

Assuming the question only expects answers to the nearest minute then 3:26 should be sufficient for the start time and 5:17 for the finish time. The boy read for 1hr 51 minutes.

(The accurate answer is 3780/143=26.433567 minutes after 3 o'clock is the start time and the finish time is 2460/143=17.202797 minutes after 5 o'clock. The boy read for 1440/13 minutes=1hr 660/13=1hr 50.769231 minutes.

Here's where the figures come from. Let's suppose that x minutes after 3:25 is the true time when the hours and minutes are interchanged. And also suppose that y minutes after 5:15 is the true time, then y=5(25+x)/60=(25+x)/12 and x=(15+y)/12. This is the algebraic equivalent of the arithmetic shown earlier. We have simultaneous equations: 12x=15+y and 12y=25+x. We can write y=12x-15 and substitute: 12(12x-15)=25+x, so 144x-180=25+x, 143x=205, making x=205/143. So the number of minutes after 3 o'clock is 25+205/143=3780/143=26.43 mins approx. And y=12*205/143-15, so 15+y=2460/143=17.20 mins approx.)

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