## FORMULAE: Time and Work

### 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 man-days 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 man-days. So 40% of the work took 50 men * 50 days = 2500 man-days. Then to complete 60% work lets assume 'X' man-days. So we get 'X' as 3750 man-days. 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 man-days:
Work needs = 12 * 8 = 96 man-day or 20*10 = 200 woman-days.
But since work is same quantity we equate both:
96 man-days = 200 woman-days
1 man-day = 2.083 woman-day
Therefore in 1 day, 12 men = 12 * 2.083 = 25 women.
Hence 25+15 women = 40 women work on a job that takes 200 woman-days to complete. So to complete task we need 200 woman-days / 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)/(u-v) 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)

### 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' = 6s2 ; Volume of a cube = s3

Cylinder: Radius of base = r and height = h;

Curved surface area of cylinder = 2πrh

Total surface area = 2πrh + 2πr2

Volume of cylinder = πr2h

Cone: Radius of base = r and height = h;Slant height = l;

Curved surface area of cone = πrl

Total surface area = πrl + πr2

Volume of cone = πr2h / 3

Sphere: A sphere is a solid ball with radius 'r';

Surface area = 4πr2

Volume of a sphere = 4πr3 / 3