What time after 4 oclock will the hands of the clock be perpendicular to each other for the first time?

Option 4 : 38(2/11) past 4 o’clock

Free

10 Questions 10 Marks 8 Mins

When the hour hand and minute hand are perpendicular to each other, angle between them = 90°

The angle between the hour hand and the minute hand is computed as:

Angle = |(11/2)M – 30H|, where M = minutes & H = hours

We have, angle = 90° & H = 4

⇒ 90 = |(11/2)M – 120|

⇒ 90 + 120 = (11/2)M

⇒ M = (2/11) × 210 = 420/11 = 38(2/11) minutes

Hence, the hour hand and minute hand will be perpendicular to each other at 38(2/11) past 4 o’clock

With hundreds of Questions based on Clock and Calendar, we help you gain expertise on Logical Reasoning. All for free. Explore Testbook Learn to attain the subject expertise with us.

India’s #1 Learning Platform

Start Complete Exam Preparation

Daily Live MasterClasses

Practice Question Bank

Mock Tests & Quizzes

Get Started for Free Download App

Trusted by 3.3 Crore+ Students

Amil, Rajiv Nooh D. - November 2021

Canoy, Armand Henzkel B. - November 2015

Corres, Nikho Lawrence R. - November 2017

Dameg, Arianne Joyce A. - May 2022

Fernandez, Renz Rodney S. - May 2018

Garibay, Sirven C. - November 2016

Labrador, Archigine A. - May 2022

Lim, Jameson Glen C. - May 2022

Lucero, Jaydee N. - November 2018

Luz, Francis Arjay P. - May 2015

Mahilum, Lou Mervin Tristan P. - November 2019

Sacar, Hisham A. - May 2017

Tan, Luvincci D. - May 2016

Tarriga, Ronie II T. - May 2019

The minute hand's angular speed is 360 degrees per hour. or 6 degrees per minute The hour hand's angular speed is 360 degrees per 12 hours, which is equal to 30 degrees per hour, or 0.5 degrees per minute. The angle between the hands is either decreasing or increasing at the rate of 6-0.5 = 5.5 degrees per minute. At 5 o'clock the angle between the hands is 150 degrees. The angle between the hands is decreasing at 5.5 degrees per minute. The hands will be together when the angle between them decreases the entire 150 degrees to 0. That will be 150/5.5 = 27 3/11 minutes. [Notice that on the hands' way to being together from 5 o'clock, they were perpendicular at one instant.] Then after the hands are together, the angle between the hands begins increasing at 5.5 degrees per minute. They will be 90 degrees apart in 90/5.5 = 16 4/11 minutes. That will be the second time after five o'clock they were perpendicular. So the answer is 27 3/11 + 16 4/11 = 43 7/ll minutes. Edwin

If the minute hand has made $x$ revolutions around the clock ($60x$ minutes), the hour hand has made $\frac1{12}x$ revolutions; the difference between these two values represents an angle. Both hands start at 0 revolutions.

For (1), the relevant angle is $\frac14$ of the circle: $$x=\frac14+\frac1{12}x$$ Solving, we get $x=\frac3{11}$, which corresponds to a time of $\frac{180}{11}$ minutes after noon.

For (2) we replace $\frac14$ with 1, since the minute hand has lapped the hour hand. Then $$x=1+\frac1{12}x$$ Solving, we get $x=\frac{12}{11}$, i.e. $\frac{720}{11}$ minutes after noon.

Therefore, both your solutions are correct.