Speed ,Time & Success
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Prologue
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Speed(S)
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Time (T)
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Distance (D)
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Time & Work (In a nutshell)
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Easy Methods
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Solved Problems
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Permutations and combinations
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Prime Number & Number Sequence in
Brief
Just an aspirant like you, so do forgive
me if you find any mistakes. Chosen
Speed, Time and distance topic because it’s a favorite topic for all public
exams
Speed,
Time & Distance (STD) are always in a committed
triangle relationship
Speed
-
Speed is how fast something is going. Another way to think of this is as how
far you can go in a certain amount of time or how fast I am writing this
article (btw it took me 4 hours to complete this article). Measured as distance
traveled per unit of time.
Example: The speed of these cars is over 150 kilometers per hour (150 km/h)
Example: The speed of these cars is over 150 kilometers per hour (150 km/h)
Just a passing Reference difference between speed
and velocity
Velocity - Velocity is speed with a direction.So if something is moving at 5 km/h that is a speed.
But if you say it is moving at 5 km/h westwards that is a velocity.
If something moves backwards and forwards very fast it has a high speed, but a low (or zero) velocity
Time
– Precious thing (I guess no need to explain time)
Distance
is the space between two objects or points (So in our example distance is
starting point of car to its finish line)
Unit
of Measurement
Unit of measurement of distance = km (kilo meter) ,
m , cm
Unit of measurement of time = hour (hr), minute
(min), second (s)
Unit of measurement of speed = km/hr, m/s
Conversions
1 km = 1000 m
1 hour = 60 min
1 min = 60 sec
1 hour = 60 * 60 = 3600 sec
1 km/hr = 1000/3600
=
5/18 m/s
e.g.
For
simple calc
1 km/hr = 0.27 m/s
1 m/s = 3.6 km/h
Now let’s turn our attention from more basic
concepts to exam oriented questions and ways to solve it
Some important formulas and you can solve any
questions related to (STD)
1)
If a TRAIN
covers a certain distance at x km/ph and an equal distance at y km/hr ,the
average speed of the whole journey =
For e.g. If a
train covers Pune to Mumbai 250 km at x=50 km/hr and Mumbai to pune 250 km at
y=60 km/hr
Then average
speed = 54.54 km/hr
2)
Speed and time
are inversely proportional (when distance is constant) ⇒
Speed ยต (when
distance is constant)
This means as the speed increases the time decreases
(or the time taken is less).
3)
If the ratio of
the speeds of A and B is a : b, then the ratio of the times taken by them to
cover the same distance is b : a or :
These are the only formulas
required for STD type of problems.
Now relating this to train formulas
as for train also speed, time and distance logic will be same. The only
difference will be when a pole or a man or any “xyz object standing”.
Laymen e.g. The Monorail started in
Mumbai (from Chembur to Wadala). Imagine its length is 100 m
1)
Time taken by a train of length d1 meters
to pass a pole or standing man or a signal post is equal to the time taken by
the train to cover d1 meters.
Answer
= Time taken for the train to cover 100 meters
2)
Time taken by a train of length d1
meters to pass a stationery object (chembur station= 50 m) of length d2 meters
is the time taken by the train to cover (d1 + d2)
meters.
Answer
= Time taken for the train to cover 150 meters
3)
Suppose two trains or two objects bodies are moving in the same direction at s1
m/s and s2 m/s, where s1 > s2, then their relative
speed is = (s1 –s2) m/s.
4)
Suppose two trains or two objects bodies
are moving in opposite directions at s1 m/s and s2 m/s, then their relative
speed is = (s1+s2) m/s.
Easy
= opposites attract. And any kind of attraction is a positive thing
5) If two trains of length d1 meters and m2
meters are moving in opposite directions at s1
m/s and s2 m/s, then:
The time taken
by the trains to cross each other =
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Few
Problems with super short cut methods
1)
A
monorail covers a distance (d1) of 5 km with a speed (s1) of 4 km/hr and
next 6 km with a speed of 3 km/hr in travelling from A to B. What will be
the train’s average speed during the whole journey?
Solution = If d1=first distance=5 km
And
d2=second distance=6 km (given in above example)
And
s1=first speed=4 km/hr
And
s2=second speed=3 km/hr
Then average speed= =
Putting
the value of d1, d2, s1 and s2 from the given example in the above formula
We
get
Average
speed=
Average
speed= (11x4x3)/ (15+24)
= (11x4x3)/39
= (11x12)/39
=3.38 km/hr
Continuing
with the concept of Monorail only
2)
A
man reaches Wadala station(to board mono
rail) from home late by 30 minutes from
his scheduled time if he walks at a
speed(s1) of 5 km/h, but if he walks at a speed(speed s2) of 6 km/h, he will
reach his home 5 minutes early. What will be the distance from station to home?
Rule-
if a person reaches his destination t1 time early by walking at a speed of s1
and t2 time later by walking at a speed of s2, then the distance between both
places is
= =
Now
equate the question given above to the rule also given just above.
Now
t1=30 minutes, t2=5 minutes
And
s1=5 km/h, s2= 6 km/h
=
= 17.5 km
3)
If
a monorail does a journey in 'H' hrs, the first half at 's1' km/hr and the
second half at 's2' km/hr. The total distance covered by the car:
=
A
monorail does a journey in 10 hrs, the first half at 21 km/hr and the second
half at 24 km/hr. Find the distance?
Ans: Distance = (2 x 10 x 21 x 24) / (21+24)
= 10080 / 45
= 224 km.
Ans: Distance = (2 x 10 x 21 x 24) / (21+24)
= 10080 / 45
= 224 km.
The
whole questions on Time speed and distance and Train revolve around these basic
formulas. But practice is needed to solve it more efficiently
Permutations and
combinations
Q1)
what is combination?
We
always use the word combination in our day to day life.
e.g "The fruit salad is a
combination of apples, grapes and bananas" We don't care what
order the fruits are in, they could also be "bananas, grapes and
apples" or "grapes, apples and bananas", its the same fruit
salad.
e.g.
The G20 includes a combination of the largest developed and industrialized
countries, which make up nearly 85% of the world’s economy
If
the order doesn't matter, it is a Combination
Q2) What is permutation?
The word permutation is though less used but often
applied in our everyday life.
e.g.
"The
combination to my bank password is 472"(not really just a hypothetical
one).
Now we do care about the order. "724" would not work, nor
would "247". It has to be exactly 4-7-2.
So,
in Mathematics we use more precise language:
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So
let’s focus on permutation first (coz order do matters in life)
Permutation
A Permutation is an ordered Combination.
More easy to remember … permutation = position (p..p)
There are basically 2 types of permutation
1) Permutation with repetition
Above example – I can keep my bank password as 474 or 444 also
(note in this the order matters, but the numbers are repeated)
Technically
speaking
When we have n things to choose from ... we have n
choices each time! When choosing r of them, the permutations are:
n × n × ... (r times)
(In other words, there are n possibilities for the first choice, AND
THEN there are n possibilities for the second choice, and so on, multiplying
each time.)Which is easier to write down using an exponent of r?
n × n × ... (r times) = nr
10 × 10 × ... (3 times) = 103
= 1,000 permutations
So,
the formula is simply:
nr
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where
n is the number of things to choose from, and we choose r
of them
(Repetition allowed, order matters) |
2) Permutation without Repetition
In
this case, we have to reduce the number of available choices each time.
In
the above bank password example.
We
have to create a 3 number password out of the 10 numbers (0,1,2,3,4,5,6,7,8,9).
But remember repetition is not allowed.
So
we choose number 4 (out of ten options we have chosen one).
We
choose number 7 (as 4 is already taken and repetition is not allowed so now we
can choose a number out of nine)
We
choose number 2 (as 4 and 7 is already taken and repetition is not allowed so
now we can choose a number out of eight)
what
if we wanted to select just 3, then we have to stop the multiplying after 14.
How do we do that? There is a neat trick ... we divide by 7! ...
10 × 9 × 8 × 7 × 6 ...
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=
10 × 9 × 8 = 720
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7× 6 ...
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The
formula is written:
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where n is the number of
things to choose from, and we choose r of them
(No repetition, order matters) |
Some
examples (It’s very easy. Just figure if repetition is there or not)
1)