CAT exam preparation videos 2024 |Time & Distance clocks 1 | | Quantitative Aptitude

Summary

In this video, Reba introduces fundamental and advanced concepts of clocks, focusing on circular motion and the relationship between the hour and minute hands. She explains how the two hands, moving at different speeds, coincide at specific points. Key principles include speed ratios, time intervals, and equidistant points where the hands meet. The video also covers the angles traveled by the hour and minute hands, relative speed, and solving problems related to time gain or loss. Practical examples and a formula for calculating the angle between the hands are provided to help viewers tackle complex clock-related questions.

Takeaways

• 🕒 Introduction to clocks: The video covers basic and advanced concepts about clocks, focusing on how the hour and minute hands behave.
• ⚖️ Speed ratio of the hands: The speed ratio between the hour and minute hands of a clock is 1:12, which helps explain how they interact and meet at distinct points.
• 🔄 Equidistant points: The hour and minute hands meet at 11 distinct, equidistant points on the clock, as derived from their speed ratio.
• 🕐 Meeting interval: The hands coincide roughly every 65 5/11 minutes, calculated based on the 12-hour system.
• 🕔 Gap between meetings: Each gap between meetings of the hands is calculated as 65 5/11 minutes, meaning that over 12 hours, they meet at 11 distinct intervals.
• 📏 Speed of hands: The hour hand moves at 0.5 degrees per minute, while the minute hand moves at 6 degrees per minute, with a relative speed of 5.5 degrees per minute.
• ⏲️ Time gain/loss in faulty clocks: A clock gaining time causes the hands to meet faster. In this example, after every 64 minutes, the clock gains 16/11 minutes.
• 🔧 Formula for angle: The angle between the hour and minute hands can be calculated using the formula: θ = 30H - 11/2M.
• 📐 Example calculations: For a time of 7:30, the angle between the hands is calculated by subtracting the total angle traveled by each hand, yielding 45 degrees.
• ⏳ Advanced clocks: The concepts extend to other types of clocks (e.g., 24-hour clocks), where the derivations and principles still apply.

Q & A

What is the concept of circular tracks discussed in the video?

In circular tracks, when two people are running in the same direction with a speed ratio of m:n (in lowest terms), they will meet at m-n distinct points on the track. These points will be equidistant.

What is the speed ratio between the hour hand and the minute hand on a clock?

The speed ratio between the hour hand and the minute hand on a clock is 1:12. This means the minute hand moves 12 times faster than the hour hand.

How many distinct points do the hour and minute hands meet on a clock?

The hour and minute hands meet at 11 distinct points on the clock.

What is the interval between two consecutive meetings of the hour and minute hands on a clock?

The interval between two consecutive meetings of the hour and minute hands on a clock is 65 5/11 minutes.

How is the angle swept by the hour hand calculated?

The hour hand moves 360 degrees in 12 hours, which means it moves 30 degrees per hour. Therefore, the speed of the hour hand is 0.5 degrees per minute.

How is the angle swept by the minute hand calculated?

The minute hand moves 360 degrees in 60 minutes, which means it moves 6 degrees per minute.

What is the relative speed between the hour and minute hands?

The relative speed between the hour and minute hands is 5.5 degrees per minute (6 degrees per minute for the minute hand minus 0.5 degrees per minute for the hour hand).

How often do the hour and minute hands coincide in a normal clock?

In a normal clock, the hour and minute hands coincide after every 65 5/11 minutes.

How much time does a clock gain if the hour and minute hands coincide every 64 minutes?

If the hour and minute hands coincide every 64 minutes, the clock gains 16/11 minutes every 64 minutes. In a day, the clock will gain 32 8/11 minutes.

What is the general formula to calculate the angle between the hour and minute hands at any given time?

The general formula to calculate the angle between the hour and minute hands is θ = |30H - 11/2M|, where H is the hour and M is the minute.

Outlines

00:00

🕰️ Introduction to Clocks and Circular Tracks

The video begins with an introduction to basic and advanced concepts related to clocks and circular tracks. It explains the fundamental idea of how two objects moving in the same direction with different speed ratios (M:N) will meet at distinct points on a circular track, with those points being equidistant. The analogy is applied to the hands of a clock, where the hour and minute hands move at a speed ratio of 1:12 and meet at 11 equidistant points on the clock.

05:01

⏳ Calculating Time Intervals Between Coinciding Hands

This section details how the hour and minute hands of a clock coincide every 65 5/11 minutes, and the interval between these coincidences is 12 by 11 hours, which equals approximately 65 5/11 minutes. It explains how the equidistant points where the hands meet are spaced evenly in a circular arrangement and highlights the mathematical calculation behind the time intervals.

10:03

🕒 Gaining Time and Clock Precision

Here, the script discusses an imaginary clock where the hour and minute hands coincide after 64 minutes instead of the usual 65 5/11 minutes, indicating that the clock is running fast and gaining time. The section calculates the difference in time gained per day, showing how to compute this gain using formulas based on the clock's faster speed.

15:07

📐 Understanding Hand Movements and Angles

This paragraph breaks down how to calculate the angle between the hour and minute hands of a clock at any given time. It introduces a formula based on the speed of the hands, with the hour hand moving at 30 degrees per hour and the minute hand at 6 degrees per minute. The relative speed between the hands is used to calculate angles and solve for their positions at specific times.

20:08

🔢 Applying the Angle Formula to Real Scenarios

The final section demonstrates how to apply the angle formula to real-world examples, such as finding the angle between the hour and minute hands at specific times like 7:30 or 10:45. It generalizes the formula for different time intervals and explores scenarios where the hands move in unusual patterns, like in clocks with 24 units or semicircular designs.

Keywords

💡Clock

A clock is a device used to measure and indicate time. In this video, the concept of clocks is central as it explains the relationship between the hour and minute hands, their speeds, and the points at which they coincide. The example of a standard clock divided into 12 parts is used to illustrate how the hands move in relation to each other.

💡Hour hand

The hour hand is the slower-moving hand on a clock, which indicates the current hour. In this video, the hour hand moves at 1/12th the speed of the minute hand, and its movement is discussed in terms of degrees per hour (30 degrees per hour). It is critical to understanding the calculation of angles and coincidence points on the clock.

💡Minute hand

The minute hand moves faster than the hour hand and completes a full revolution (360 degrees) every 60 minutes. The video explains that the speed of the minute hand is 6 degrees per minute and compares its movement with that of the hour hand, particularly how they coincide every 65 5/11 minutes.

💡Speed ratio

The speed ratio refers to the rate at which the hour hand and minute hand move relative to each other. In the video, this ratio is 1:12, meaning the minute hand moves 12 times faster than the hour hand. The concept is used to calculate the time intervals at which the two hands coincide on a circular track.

💡Coincidence points

Coincidence points are the moments when the hour and minute hands overlap or coincide at the same position on the clock. The video calculates that these overlaps occur at 11 distinct points on a 12-hour clock, with each point being equidistant from the others. The time interval between coincidences is approximately 65 5/11 minutes.

💡Equidistant points

Equidistant points refer to points that are evenly spaced along the circular path of the clock. In the video, it is explained that the hour and minute hands meet at 11 equidistant points on the clock over a 12-hour period. These points are crucial for understanding the predictable intervals between their meetings.

💡Relative speed

Relative speed is the difference in speed between two moving objects—in this case, the hour and minute hands. The video explains that the relative speed of the hands is 5.5 degrees per minute (6 degrees per minute for the minute hand minus 0.5 degrees per minute for the hour hand). This value is used to calculate the time it takes for the hands to coincide.

💡Angle between hands

The angle between the hour and minute hands at a given time is a key concept in the video. It is calculated by determining how far each hand has moved from the 12 o'clock position. The video provides formulas for calculating this angle based on the current time and the relative speeds of the hands.

💡12 by 11 hours

This refers to the division of 12 hours by 11 to find the interval between coincidence points of the hour and minute hands. The video shows that the hands coincide every 12/11 hours, or approximately 65 5/11 minutes. This time interval is fundamental for predicting when the hands will overlap.

💡Clock gain or loss

Clock gain or loss refers to how much a clock deviates from standard time. The video provides an example of a clock where the hour and minute hands coincide after 64 minutes instead of the normal 65 5/11 minutes, indicating the clock is running fast. It shows how to calculate how much time the clock gains over a day due to this difference.

Highlights

Introduction to the concept of clocks and circular tracks.

Explanation of how two people running in the same direction with different speed ratios meet at equidistant points on a circular track.

Clarification that the speed ratio of the hour hand to the minute hand on a clock is 1:12.

Derivation that the hour and minute hands meet at 11 distinct equidistant points on a clock face in 12 hours.

Calculation of the interval between these meetings, resulting in 65 5/11 minutes.

Explanation of the angular speed of the hour hand as 30 degrees per hour or 1/2 degree per minute.

Explanation of the angular speed of the minute hand as 6 degrees per minute.

Derivation of the relative speed between the hour hand and the minute hand as 5 1/2 degrees per minute.

Problem-solving example: The hour and minute hands coincide every 64 minutes in an imaginary clock, indicating the clock is running fast.

Explanation of how to calculate time gained by the fast clock in one day, resulting in 32 8/11 minutes.

Introduction of a formula to calculate the angle between the hour and minute hands at a given time: θ = 30H - 11/2M.

Example: Calculation of the angle between the hour and minute hands at 7:30, resulting in 45 degrees.

Explanation of how the formula applies to different types of clocks, including ones with 24 divisions.

Example: Calculation of the angle between the hour and minute hands at 10:45, resulting in 52.5 degrees.

Example: Calculation of the angle between the hour and minute hands at 5:48, resulting in 104 degrees.

Transcripts

[Music]

hi everyone my name is Reba - and

welcome with a first class of Crocs so

we look a look at all the basic concepts

of clocks and we do some good advanced

questions also in blocks okay so see ya

like suppose the first thing is if I

draw a clock here like 12 6 3 & 9 so for

the concept of circular is a circular

traps we know that whenever two people

are running in the same direction

and then speed ratio is M h2n whenever

two people are running in same direction

and then speed ratio as a mesh to n they

will meet at M - n distinct the points

on the clock or on the track points and

those points will be equidistant right

this we know by circular tracks right so

whenever two people are running in same

direction in the ratio of speed ratio of

M H to n where m is 2 n is in lowest

form right lowest form after canceling

raw edge form then their meet at M - n

distinct points on the clock and those

points will be equidistant on this track

so your clock track okay then we get n

minus n distinct parts on the track

circular track and those points will be

equidistant so same way in clock what

happens we have to be burning also here

our handed minute hand okay so our hand

and minute hand are running in clock and

speed ratio here is then a speed ratio

here is one is to twelve one is to

twelve right so speed of our hand and

minute hand in a clock is in the ratio 1

H 2 12 right why so because suppose our

hand and minute hand are at this point

thumb and exactly at 12 so in a timely

so now let us say clock is divided into

twelve parts right one two three legged

clock

divided into 12 parts so when this our

hand covers one part of clock when this

our hand covers one part of cloth in

that time this minute hand comes from

here to here color into n parts of drum

so when our hand moves one part minute

and both 12 part that when the disparate

ratio is 1 1 H 2 12 okay so that means

if there is predacious 1 to 12 and then

running in same reaction obviously our

hundred million minute in that movie

same direction right same direction only

right it's only in horror movies that

they move in upward direction okay

normally they move in same direction

okay so other hand will attend the speed

ratio so I should well that missile me

dad

they'll meet at 11 distinct points on

the clock they'll meet at 11 distinct

points on the clock right so ratio here

is MH 2 N and they will meet at 11

distinct points on the clock because M

minus n distinct points towards minus

111 distinct points right that basically

means that if I make a clock here so

they will be meeting at 11 equidistant

points on the clock so it's the first

time they are at 11 together okay 11

they are together then after what time

again they'll meet together right

roughly what again say roughly so they

will meet at roughly 1/5 1/5 our

American almost there

add to 10 power and wait and almost

there right

so what it exact time unit so this is 11

distinct points 1 2 could be 3 right

then for like that so 11 distinct point

11 15 points that are equidistant the

gap is equal this gap is equal right

this gap is equal equal gap because

they're equidistant points a gap is

equal okay that mystery means that so in

full 12 hours right for whole 12 hours

we we take 11 distinct points in 11

dating

so that means 11:15 point at wizard 11

gaps also here right there are each gap

is equal to the element because it's

circular dragna so number of points is

equal to number of gaps also right if it

is linear then 11 points your draw you

will get 12 gaps right but here it is

circulant so 11 points 11 gap right so

xi equal gaps consists of 12 hours I get

thing like that

so xi equal gaps consists of 12 hours so

what is what is the what is the interval

of 1 gap so interval of 1 gap is equal

to 12 by 11 hours 12 by 11 hours is

equal to 12 by 11 into 60 minutes this

is equal to 720 by 11 minutes type so

therefore now 720 by referee to solve we

will get 55 so you get 65 5 by 11

65 5 by 11 minutes it is a very point a

time right what is data is this data is

that after every 65 5 by 11 minutes on a

clock 11 minutes in a clock the hour

hand and minute hand will coincide with

each other will meet all I can say

coincide with each other right so it's a

logic behind it right it's a logic

behind it is again same thing when

they're if they're moving in the ratio

of Rho tools then we had 11 distinct

points on the clock those 11 15 points

are equidistant but that means air gap

is equal what is it interval of one gap

so total 11 gap is consisting of 12

hours so what is it what is the gap what

is the value of one gap 12 by 11 hours

that is 65 5 by

that means this gap is also this gap

also 65 5 by ll minutes this gap also 65

5 by elements so all the gaps equal to

65 5 by element right so this is a very

own data in clock okay then we will see

what is the and this wrapped by hour and

minute and right what is the angle swept

so our hand and minute hand okay so for

our hand what happens when fertility

speed ratio right what is the East Bay

detainee look at the Honda also today is

field of our hand so in a clock when our

hand will move from 12 to 12 other hand

will curl covered 12 parts of clock and

total 360 degree right so 360 degree it

covers in 12 hours our hand okay that

means what is the speed here it's a

speed is 30 degree per hour so spirit of

our hand is in terms of our is 30 degree

per hour

in one hour it moves 30 degree into also

2 degree like that okay then if I do if

I say minutes

so ok then all 30 degree that is per 60

minutes so in every 60 minutes 30 degree

that means it should be 1/2 degree 1/2

degree per minute it should be 1/2

degree per minute this is a speed of our

video right so our hand is moving at a

rate of 1/2 degree per minute or 30

degree per hour by order it minute hand

is moving minute and if you see minute

hand in the same clock so minute hand

moves like this right suppose is a

minute hundred twelve right now okay

so minute hand in 60 minutes covers full

360 degree right so minute hand in 60

minutes rejects from - a little - well

that is whenever

did you reach us from 12 to 12 so minute

hand covers full 360 degrees 60 minutes

so 360 degree in 16

that word is values what six degree per

minute so the speed of minute hand is

what six degree per minute in speed of

our and is what half degree per minute

okay and what is that a relative speed

relative speed of our hand and minute

hand his word since both are voice

erections of different speed that is six

minus half that is five and half degree

per minute so we will use all these

concepts for solving the questions we

will use all these concepts when solving

the questions okay okay here's a

question here so our hand and minute

hand of an imaginary clock coincides

after every 64 minutes how much time the

clock will gain or lose in a day right

so what we know now as in any normal

clock right as in a normal clock any

normal clock our hand and minute hand

coincides after every sixty five five by

eleven minutes this happens a normal

clock right we just restricted so but in

this clock but in this clock there

co-signing after sixty four minutes

those two hands are to meet after

traveling for sixty five-plus minutes

right and in this case they are waiting

at only sixty four minutes that means

hands are moving fast right if hands are

moving fast

that means the clock is gaining time not

losing time if hands are moving fast

that means clock is gaining time right

summer time clock has gained so I can

say that after traveling for after

traveling for 64 minutes how much time

they have gained they have gained the

difference for a difference here from 65

five by 11 their differences of 1 5 by

11 minutes

so after traveling for 64 minutes they

have gained one fight by ll minutes

right that is equal to 16 by ll minutes

this offender ID so after traveling for

64 minutes how much time they have

gained so they were supposed to meet at

65 five by the ll minutes right

but in this club they are meeting at

only after 64 minutes that means the

clock has gained 16 by Ln minutes after

which 64 minutes so they are not simple

right so in 64 minutes they are gaining

16 by 11 minutes they are giving so when

your entry method in one minute they

will gain how much 16 by 11 into 64

minutes they will gain right this is a

basic thing here this in pointing

calculate for one minute

read the question tell me anything right

how much time it goes in a day in our

you know in 12 hours in 8 hours right

that simple thing but the main thing is

to calculate till this part so in 1

minute the clock will lose how much 16

by 11 in - sorry clock will gain how

much 16 by 11 in 264 minutes this first

time the clock will gain okay so now

question is in a day how many to lose

gain in gain in a day how much so you

know Damien's 24 hours so in 1 minute

how much in one went on in this one so

in 60 minutes it should be 16 by 11 into

64 into 60

so therefore in a day in one day that

means in 24 hours and multiplied by 24

16 by 11 into 64 into 60 into 24 this

should be the answer right if you solve

it utility value as 360 by 11 that is

equal to 32 8 by 11 minutes so the

answer this question will be 32 8/2 by

11 minutes this first time the clock

will gain in a day this much times it

will often get in a day if the hour and

minute hand call sides often set if

it be

so now is what we'll do we'll try to

find the angle between our hand and

minute hand in a clock right so it is

important here is see once you're trying

to find the angle between hour and

minute hands so in a general flow this

atomic normal formula right we'll try

that formula and remember it okay but

the thing is that derivation is very

important because in exams I don't

expect much questions on normal clock

right so what their questions are giving

this it is a different block like there

are 24 minutes on the clock

instead of 12 or there is a semicircle

kind of cloth right so in those

questions that I have to give her to

find the angles funda right then you

should know this how - how we have

derive this that derivation will be

applicable to that the different blocks

also right but but that formula will not

be applicable just formulate only for

our clock this general clock when there

are 12 units on a clock

otherwise this formula is never ever

given so that's I devised this formula

this very important deliveries are

important for the other other questions

point of view and this force simply

knievel the formula for normal clocks

right so you derive it now find the

angle between our certain time so let's

say this is 12 9 6 3 it's same time is

down for example take any time it's the

time is now 730 kind of okay so minute

hand is here at 6:00 hour and rewritten

7 so we're trying to drag this angle

here this and this theta I am trying to

today right always my paces I do it so

can you say let's assume that this 12

the hour and minute hand wasn't at 12

only right so initially our 100 minute

and both are at will and after that is

7:30 so a logic will be how to find this

theta this theta will find by this total

until traveled by our hand total and it

traveled by our hand - the angle

traveled by minute hand all right so in

logic will be here the total and

they traveled by our hand from 12 okay

the total angle traveled by the total

angle traveled by our hand till - well -

the angle traveled by minute hand this

will give me this theta this is my logic

here okay so how to approach how to

approach square simple so we just write

how to target theta in any clock right

in any club theta is equal to and then

traveled by our hand - and well traveled

by minute hand so obviously you can say

that with minute hand is above a eight

minute hand is above our hand in that

case and we'll travel to a minute hand

suppose minute hand was here supposed

minute hand was here in this case and

we'll provide by minute hand

- angle traveled by our hand right left

reverse it so it's okay you don't need

to take that into consideration here

right

so and give the road by our hand - I

control a minute

what is handled by our and here so we

know that angle our a travels 30 degree

per hour we have done this

so our and travel 30 degree per hour so

till 7:00 so next time is 7:30 now so

till 7:00 the our hand travels 30 degree

for seven hours each house right 30

degree for each hour so for seven hours

thirteen - so - right now all and is

here he's had cleared seven right but it

has to go at 7:30 that is for extra 30

minutes the our editor how much now what

what is the animal turban so our

intervals at half degree per minute

so in 30 minutes our and moveset half

degree into 30 30 minutes right seven is

our 30 is minutes right and then we'll

generalize for H hours and a minute

okay this one seven and thirty seven

thirty so this total angle traveled by

hour and 30 degree in for seven hours

from here to here 30 degree for seven

hours

and blessed for this extra 30 minutes

huh that is half daily per minute that

is half into 30 now - and they throw

away minute handed so the Godman had

been here and minute hand after day by

day or so minute hand has prevailed from

12 to 6 that is for 30 minutes so minute

hadtravel sexy degree per minute so in

30 minutes

minute handle travel 6 degree into 30

minutes per minute rate this will be the

answer to to 10 plus 15 to 25 minus 180

- ever been 45 degree should be my

answer 45 degree correct

so now generates for the generalizes for

eight hours and ten minutes so thirty

into seven is h-here hour plus half into

em 30s minutes here - six again dis

degree and 30 is minutes is 6m so what I

am getting here is at HM that we said H

hours and minutes theta is equal to 30 h

minus 11 by 2 and this is the formula i

got for general clock right but this

television is very fun for any other

clock if it comes in the question like

the clock is divided into 24 parts or

thirty six parts o'clock instead shape

of a quarter circle or semi circle like

that

this is very useful that derivation

again shadow do like that so theta is

equal to 30 h minus Ln my to n when when

h is above m

minute hundred at six H is here hour and

is here right if minute hand rose above

our and in that case it should be

ordinary worship 11 by 2 m minus 30 h 11

by 2 m minus 30 gauge right this is a

formula in this case okay so just for

example okay suppose their time is now

1045 so this theta

1045 so theta that is that what is the

angle between our and hundred minute and

at 10:45

so our hand is basically H is 10 and M

is 45 so they simply see that other hand

is above minute hand right because our a

digital plus and made at any other time

what is the time now this time like this

I'd simply browse it however it is here

and minute hand is here right at 10:45

that means our hand is above minute hand

so use the formula 30 H that means our

hand - minute hand so usually formula

30h - or an input mode also but it's

fine

it's a one-second process right no no

not even one second if H is above our

hand use H minus M if M is above our

hand use em - stitch that's same thing

so H is our above minute hand our head

is ever been at hand use H minus m hu

replaced by 10m you replace by 45 so you

get three hundred minus twenty two point

five into eleven so 300 - or twenty two

point five to eleven is to forty seven

point five that is equal to fifty two

point five degrees since the answer if

the question was what is your typing at

5:48 so for 548 is a time 5:48 p.m. you

let us say what is the value of theta

here so at 5:48 in the situation other

hand is here and minute hand is here so

minute hand is above our hand not with

reference from from 12 always right so

minute our hand if they were less

distance minute hand have driven more

different distance in that case it

should have minute minute hand - our

hand at miss 11 by 2 m minus 30 H that

is 11 by 2 into 48 minus 30 into 5 that

is equal to 104 degrees in 24 into

eleven to 64 minus one foot point 51 114

that is the answer right there is the

most busy part of clock right so you

should know all this to sort of are

questions of drop so in the next videos

we'll see how to solve tougher pushers

of clock okay introduce a square using

these concepts we see different types of

clock also in clock you can see myself

that or water settle all different kind

of how to solve it ok thanks for

watching

you