Aptitude Discussion

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

✖ A. |
22 |

✔ B. |
22 |

✖ C. |
23 |

✖ D. |
24 |

**Solution:**

Option(**B**) is correct

Eleven times in 12 hrs. The hands will be directly opposite each other at the following times:

12:32 and 8/11 of a minute

1:38 and 2/11 of a minute

2:43 and 7/11 of a minute

3:49 and 1/11 of a minute

4:54 and 6/11 of a minute

6:00

7:05 and 5/11 of a minute

8:10 and 10/11 of a minute

9:16 and 4/11 of a minute

10:21 and 9/11 of a minute

11:27 and 3/11 of a minute.

**Therefore, the hands of the clock will be opposite to each other 22 times.**

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

**Robin**

*()
*

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,

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

So, $180^\circ$ angle (event of opposition) will be obtained $\textbf{= 22 times.}$