Aptitude Discussion

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

✖ A. |
21 |

✔ B. |
22 |

✖ C. |
23 |

✖ D. |
24 |

**Solution:**

Option(**B**) is correct

The hands of a clock coincide 11 times in every 12 hours (Since between 11 and 1, they coincide only once, *i.e.,* at 12 o'clock).

**AM**

12:00

1:05

2:11

3:16

4:22

5:27

6:33

7:38

8:44

9:49

10:55

**PM**

12:00

1:05

2:11

3:16

4:22

5:27

6:33

7:38

8:44

9:49

10:55

The hands overlap about every 65 minutes, not every 60 minutes.

**Thus the minute hand and the hour hand coincide 22 times in a day.**

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

**Asfandyar**

*()
*

**Kapil Shukla**

*()
*

Total minutes in a day =$24x60$

Hands overlap in a day= after every $65$ minutes not $60$ minutes

Then both hand coincide in a day = $24x60/65= 22.22222....$

This can not be decimal therefore answer is = $22 $times

**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,

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

So, $0^\circ$ angle (event of coincidence) will be obtained $\textbf{= 22 times.}$

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

For hours hand:

a). 24hours -------------------> 2 x 360= 720 degrees (Hours hand complete 2 full rotations in a day)

b). 24 x 60 (min) --------------> 720 deg

c). 1 min ------------------------> 720/(24 x 360) = 720/1440 = 1/2 degree.

Hope it helps