CAT exam preparation videos 2024 |Time & Distance clocks 1 | | Quantitative Aptitude
Summary
TLDRIn 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
🕰️ 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.
⏳ 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.
🕒 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.
📐 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.
🔢 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.
Mindmap
Keywords
💡Clock
💡Hour hand
💡Minute hand
💡Speed ratio
💡Coincidence points
💡Equidistant points
💡Relative speed
💡Angle between hands
💡12 by 11 hours
💡Clock gain or loss
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
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