
FORMULAE
Time and Work
Calculation of work done in a time:
If A does a task in 3 days and B does a task in 4 days. The in
one day A does 1/3 work and B does 1/4 work.
Both working together do: ( 1 / 3 ) + ( 1 / 4 ) = ( 7 /
12 ) work. Thus the work requires 12 / 7 days to be completed
when both work together.
Calculation of negative work done in a time:
If A builds a wall in 3 days and B builds a wall in
4 days, C breaks a wall in 12 days. The in one day A does 1/3
work and B does 1/4 work.
Total work done in a day = ( 1 / 3 ) + ( 1 / 4 )  ( 1 / 12 )
= 6/12 = 1/2
Total days to build a wall = 2 days.
Work equivalence method / Calculation to solve time
and work problems:
In such problems, concept of mandays is
important. Example: If a contractor hires 50 workers to
complete a road in 100 days. Then he finds out that in 50 days
only 40% of road building is done. So how many more days shall
he need to complete the road. How many more men would he need
to build his road on time.
Case  I
Here 40% work is done in 50 days so 60% work shall be
completed in 'x' days. By cross multiplying we get x as 75
days.
Case  II
But if work has to be completed on time then find the
mandays. So 40% of the work took 50 men * 50 days = 2500
mandays. Then to complete 60% work lets assume 'X' mandays.
So we get 'X' as 3750 mandays. But days are fixed as 50 so
3750 / 50 = 75 men are needed to finish job on time.
Equating men, women and work:
Suppose 8 men can do a job in 12 days and 20 women can do a
job in 10 days. In how many days can 12 men and 15 women do
the job.
For solving this again we need concept of mandays:
Work needs = 12 * 8 = 96 manday or 20*10 = 200 womandays.
But since work is same quantity we equate both:
96 mandays = 200 womandays
1 manday = 2.083 womanday
Therefore in 1 day, 12 men = 12 * 2.083 = 25 women.
Hence 25+15 women = 40 women work on a job that takes 200
womandays to complete. So to complete task we need 200
womandays / 40 woman = 5 days.
Problems on Trains and Poles
Calculating time to pass a pole or man who is
stationary:
Time taken by a train of length X metres to pass a pole or a
standing man or a signal post is equal to the time taken
by the train to cover X metres.
Calculating time to pass a object that has a width:
Time taken by a train of length X metres to pass an
object of width Y meters is equal to the time taken by the
train to cover X + Y metres.
Calculating time to pass a train moving in same
direction:
Suppose two trains or two bodies are moving in the
same direction at A m / s and B m/s, where A > B, then
their relatives speed = (A  B) m / s.
If two trains of length a metres and b metres are moving
in the same direction at u m / s and v m /
s, then the time taken by the
faster train to cross the slower
train = (a+b)/(uv) sec.
Calculating time to pass a train
moving in opposite direction:
Suppose two trains or two bodies are moving in
opposite directions at A m / s and B m/s, then their relative
speed is = (A + B) m/s.
If two trains of length a metres and b metres are moving in
opposite directions at u m / s and v m/s, then time
taken by the trains to cross each other = (a + b)/(u+v)
sec.
Calculating time to reach respective destinations after
crossing each other:
If two trains (or bodies) start at the same time from points A
and B towards each other and after
crossing they take a and b
sec in reaching B and A respectively,
then
(A's speed ) : (B’s speed) = (b^^{1/2}: a^^{1/2})
Practice Exercise:
Time, Speed, Distance

Click
here

Geometry
Calculate area of an equilateral triangle = (
√3 * a ) / 2
( a= length of side)
Pythagoras theorem: ( hypotenuse )^{2}
= ( side1 ) ^{2} + ( side2 ) ^{2}
^{
}
Basic Pythagoras triplets: If side dimensions are given
then hypotenuse can be directly written from the triplets:
3,4,5; 5,12,13; 7,24,25; 8,15,17; 9,40,41; 11,60,61;
12,35,37; 16,63,65; 20,21,29; 28,45,53;^{
}
^{
}
Cuboid: If length = l ; breadth = b; height =
h; Then surface area of cuboid = 2 * (lb + bh + lh); Volume
= lbh;
^{
}
Cube: Surface area of a cube with side 's' =
6s^{2 }; Volume of a cube = s^{3}
Cylinder: Radius of base = r and height = h;
Curved surface area of cylinder = 2πrh
Total surface area = 2πrh
+ 2πr^{2}
Volume of cylinder = πr^{2}h
Cone: Radius of base = r and height = h;Slant
height = l;
Curved surface area of cone = πrl
Total surface area = πrl
+ πr^{2}
Volume of cone = πr^{2}h
/ 3
Sphere: A sphere is a solid ball
with radius 'r';
Surface area = 4πr^{2}
Volume of a sphere = 4πr^{3}
/ 3