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How many times do the hands of a clock make an angle of \[{90^\circ}\] in \[36\] hours?
A) \[11\]
B) 22
C) 44
D) 72
E) 66

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Last updated date: 20th Jun 2024
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Answer
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Hint: We can solve these types of the question by using some properties of the clock which states that the minute hand of the clock overtakes the hour hand twice in one hour to give an angle of \[{90^\circ}\].

Complete step-by-step solution:
Step 1: First of all, we will draw a clock for a better understanding of the clock properties:

Step 2: Now, we will suppose that both the hour and minute hand is on twelve. When the minute hand will come to a point at three and nine, it will make an angle of \[{90^\circ}\] with the hour hand. So, we can say that in one hour the minute hand and hour hand overtakes itself twice to give an angle of \[{90^\circ}\].
Similarly, for the next hour, again it will form the right angle two times. This property will repeat till both the hour and minute hand again comes at point twelve.
So, the total number of \[{90^\circ}\] angles in the \[12\] hours of a day equals to \[11 \times 2\], because twelve repeats itself two times in these \[12\] hours.
\[ \Rightarrow {\text{Total number of }}{90^\circ}{\text{angle in }}12{\text{ hours}} = 22\]
Step 3: Now, by repeating step number 2, for the next \[12\] hours, the total number of \[{90^\circ}\] angles will be equals to \[11 \times 2\] as shown below:
\[ \Rightarrow {\text{Total number of }}{90^\circ}{\text{angle in next }}12{\text{ hours}} = 22\]
Step 4: Again, by repeating step number 3, for the next \[12\] hours, the total number of \[{90^\circ}\] angles will be equals to \[11 \times 2\] as shown below:
\[ \Rightarrow {\text{Total number of }}{90^\circ}{\text{angle in next }}12{\text{ hours}} = 22\]
Step 5: Thus, the total number of \[{90^\circ}\] angles in \[36\] hours will be equals to as below:
\[ \Rightarrow {\text{Total number of }}{90^\circ}{\text{angle in 36 hours}} = 22 + 22 + 22\]
By doing the addition into the RHS side, we get:
\[ \Rightarrow {\text{Total number of }}{90^\circ}{\text{angle in 36 hours}} = 66\]

Option E is the correct answer.

Note: Students need to take care while calculating the number of times \[{90^\circ}\] angle comes in \[12\] hours, students generally multiply two with \[12\] instead of \[11\]. You should remember that the $12^{th}$ hour of the day repeats itself twice.