Question

What is the angle between the hands of a clock when it shows \[9:45\] ?
(a) \[{\left( {7.5} \right)^ \circ }\]
(b) \[{\left( {12.5} \right)^ \circ }\]
(c) \[{\left( {17.5} \right)^ \circ }\]
(d) \[{\left( {22.5} \right)^ \circ }\]

Answer 1

Hint: The numbers on the surface of any clock form a circle so the center of the clock makes an angle of \[{360^ \circ }\]. The angle subtended at the center of the clock by any two consecutive numbers is \[{30^ \circ }\]. At \[9:45\], the hour hand is almost towards \[10\] and the minute hand is at \[45\]. So we have to find the angle between them. To do so use the formula \[\theta = \dfrac{{11m}}{2} - 30h\], where \[h\] is the hour, \[m\] is the minute and \[\theta \] is the angle between the hour hand and the minute hand.

Complete step by step solution:
First try to visualize a clock, the numbers are arranged in any clock to form a circle. So any clock makes an angle of \[{360^ \circ }\] at the center. Now there are \[12\] numbers in a clock from \[1\] to \[12\]. Any two consecutive numbers subtends an angle at the center. The total number of such angles is \[12\]. So the value of each such angle will be:
\[\dfrac{{{{360}^ \circ }}}{{12}}\] \[ = \] \[{30^ \circ }\].
Now we need to find the angle between the hour hand and the minute hand when the clock reads \[9:45\]. Observe carefully that at \[9:45\] the hour hand will be almost near \[10\] and the minute hand will be at \[9\].
To find this angle use the formula :
\[\theta = \dfrac{{11m}}{2} - 30h\]
 where \[h\] is the hour, \[m\] is the minute and \[\theta \] is the angle between the hour hand and the minute hand.
Given time is \[9:45\], so here \[h = 9,m = 45\]:
Substituting these values in the formula we get,
\[\theta = \dfrac{{11 \times 45}}{2} - \left( {30 \times 9} \right)\]
\[ \Rightarrow \theta = \dfrac{{495}}{2} - \left( {270} \right)\]
\[ \Rightarrow \theta = 247.5 - 270\]
\[ \Rightarrow \theta = - {22.5^ \circ }\]
Hence the angle between the two hands is \[{22.5^ \circ }\][ here \[ - \] indicates the direction, that is hour hand is to the left of minute hand by \[{22.5^ \circ }\]]

So the correct option is (d)

Note:
Students must be very careful that although the time is \[9:45\], the hour hand is not exactly at \[9\] and so the angle between the hour and minute hand will not be \[{0^ \circ }\]. The problem can also be solved as follows:
The hour hand travels \[{30^ \circ }\] every hour. So for \[60\] minutes it travels \[{30^ \circ }\], so for \[1\] minute it will travel \[{0.5^ \circ }\]. So in \[45\] minutes, it will travel \[{\left( {45 \times 0.5} \right)^ \circ } = {22.5^ \circ }\].