8. Priti is learning a new language called Tedio. During her one hour lesson, which started at midday, she looks at the clock and notices that the hour hand and the minute hand make exactly the same angle with the vertical, as shown in the diagram. How many whole seconds remain until the end of the lesson?
8. 276 Let the angle that each hand makes with the vertical be $x$ degrees and let the current time be $s$ seconds after midday.
In one complete hour the hour hand will turn $30^{\circ}$. There are $60 \times 60=3600$ seconds in an hour so it takes $3600 \div 30=120$ seconds for the hour hand to turn one degree.
Therefore $s=120 x$.
In one complete hour the minute hand will turn $360^{\circ}$. There are 3600 seconds in an hour so it takes $3600 \div 360=10$ seconds for the minute hand to turn $1^{\circ}$. But the minute hand has turned clockwise through an angle of $(360-x)^{\circ}$. Therefore $s=10(360-x)$. Equating the two expressions we have obtained for $s$ we obtain the equation $120 x=10(360-x)$. The solution to this equation is $x=\frac{360}{13}$. Therefore the number of seconds elapsed since midday is $120 \times \frac{360}{13}=\frac{43200}{13}=3323 \frac{1}{13}$. The number of whole seconds remaining is $3600-3324=276$.