Aptitude Discussion

Q. |
How many times will minute hand and hour hand perpendicular in one day? |

✖ A. |
22 |

✖ B. |
23 |

✖ C. |
24 |

✔ D. |
44 |

**Solution:**

Option(**D**) is correct

In 12 hrs. the hands of the clock will be perpendicular to each other for 22 times as in 1 hr. time the hands of the clock wil be at rt. angles for 2 times but between 1 1'o clock and 1'o clock they will be at rt. angles only twice.

**Therefore, in 24 hrs. they will be at right angles 44 times.**

**Edit:** Thank you **Robin** for providing an alternative solution in the comments.

**Akshay**

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**Somshekhar**

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There is a mistake in robin soluion that in hour case it is 720/1440 not 1440/720.

**Bharat**

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This solution is good but there is one mistake.

It should be :

The hour hand covers 2 X 360 in 240 i.e. 1440 minutes. So, in one minute it cover 720/1440= 1/2 degree

**Robin**

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Here is another method to solve the problem:

The minute hand travels an angular distance of 360 degrees in 60 minutes. So, it covers 6 degrees in one minute.

The hour hand covers $2\times 360^\circ$ in 24 hours, i.e 1440 minutes. So, in one minute, it covers $\dfrac{1440}{720} = \left(\dfrac{1}{2}\right)^\circ$.

Difference in angular distance travelled by the minute hand and the hour hand in one minute is thus $6-\dfrac{1}{2} = \dfrac{11}{2}$ degrees. So, on a full rotation $(360^\circ)$, any similar event between them will be repeated every $\dfrac{360}{(11/2} = 65 \dfrac{5}{11}$ minutes.

In twenty four hours, i.e. $24 \times 60 = 1440$ minutes, the number of times angle between the minute hand and hour hand will be same, can be calculated by dividing 1440 minutes with this value, thus,

$n=\dfrac{1440}{720/11}=22$

$90^\circ$ angle (event of perpendicular disposition) will be obtained $= 22\times 2 = \textbf{44 times}$ (as this situation arises twice on a complete rotation).

how hour hand covers (2*360)

as if minute hand cover 360 than hour hour should also?

is it?

please elobrate

The answer is correct but the reasoning is a bit wrong. For every hour the two hands will be a right angles 2 times except between 2 & 3 and 8 & 9 for each of which it will be only once. Between 2 and 3 the right angles appear at approx. 2:25 and 3:00. The 3:00 right angle is common for 2-3 and 3-4. Similarly between 8 and 9. So $(12*2)-2=22$ and in a day $22*2=44$