How many degrees, minutes and seconds are respectively passed over in $11\frac{1}{9}$ minutes by the hour and minute hands of a watch?

To compute the degrees of movement for the minute we need to start by figuring out what fraction of a right angle was passed over ($11\frac{1}{9}$ minutes is less than $15$ minutes or one quarter of an hour.) $11\frac{1}{9}=11.1111$ (about.)

$11.1111\times 60=666.66667$ seconds

$15\times 60=900$ seconds

$\frac{666.66667}{900}=0.740741$  (fraction of right angle passed over.)

Therefore, movement of minute hand is $66$ degrees and $40$ minutes.

The hour hand is similar, except that over a period of an hour the hour hand will only move $1/12th$ of the clock face. This corresponds to $30$ degrees. Therefore we need to compute what fraction of $60$ minutes is $11.1111$ minutes and apply that fraction to $30$ degrees.

$\frac{11.1111}{60}=0.185185$ (fraction of hour passed over.)

$30\times 0.185185=5.5556$ degrees

$60\times 0.5556=33.336$ minutes

$60times0.336=20$ seconds

Therefore, movement of hour hand is $5$ degrees, $33$ minutes and $20$ seconds.