Important formulas :
If A can do a piece of work in X days and B can do the same work in Y days, then both of them working together will do the same work in ( {xy over x + y} ) Days
A, B , C while working alone can complete a work x, y, and Z days respectively then they will together complete the work in ( { XYZ over XY + YZ + ZX } ) Days
Example 1: If man A completes a piece of work in 12 days & the same work is completed by B & C in 15 days respectively. In how many days the work is completed work together?
Solution: Work per day of A , B and C ( = {1 over 12} + {1 over 15 } + {1 over 20} )
( = { (5 + 4 + 3) over 60 } = {1 over 5 } )
Total days = 5 days (Answer)
Formula method :
They together complete the work
( = { XYZ over (XY + YZ + ZX) })
( = {12 times 15 times 20 over 12 times 15 + 15 times 20 + 20 times 12} )= 5 days (answer)
Two – person A and B working together, can complete a piece of work in X days. If A alone can complete the work in Y days, then B working will complete the work in ( {xy over x – y} )Days
Example 2 : A and B together can do a piece of work in 5 days, A alone can do it in 8 days. B alone can do the same piece of work in?
Solution:
Hence, (A + B ) = 5 days , A = 8 days
B alone can finish the work in one day
( = {1 over 5 } – {1 over 8 } )
( = {8 – 5 over 40 } )
B finish the whole work ( = { 40 over 3 } ) Days
( = 13 {1 over 3 } Days (answer) )
Formula method :
By formula ,
( = {xy over x – y } )
( = {8 times 5 over 8 – 5 } )
( = 13 { 1 over 3 } ) days (Answer)
If A and B working together, can finish a piece of work in X days, B and c in days, c and A in Z days then,
– A, B and C working together will finish the job in
( ={ 2XYZ over XY + YZ + ZX } ) Days
Example 3: A and B can complete a piece of work in 8 days 12 days B and C can do it in 8 days, A, B, and C together can complete it in?
Solution: Work done by A and B in one day ( = {1 over 8} )
Work done by B & C in one day ( = {1 over 12} )
Work done by C and A in 1 day ( = {1 over 8} )
Work done by A , B and C together in 1 day
( = {1 over 2} ( {1 over 8} + {1 over 12} + {1 over 8} )= {1 over 6 } )
A, B and C together complete the whole job = 6 days
Formula method :
A + B + C can do finish the whole work
( = { 2XYZ over XY + YZ + ZX } )
( = { 2 times 8 times 12 times 8 over 8 times 12 + 12 times 8 + 8 times 8 } )
( = { 2 times 8 times 12 times 8 over 96 + 96 + 64 } )
= 6 days ( answer )
Work efficiency of any worker inversely proportional to the time taken by him.
i.e work efficiency ( α = {1 over Time} )
Example 4: A is twice as good workman as B & together they finish a piece of work in 18 days. In how many days will A alone finish the work?
Solution: A : B = 2:1
Time is taken A and B = 18 days
Work done by A and B ( = 18 times 3 = 54 ) days
A will do alone ( = { 54 over 2 } = 27 ) days
Man – work – Hour – formula :
- More man can do more work
- More work means more times required to do work
- More man can do some work in less time
If M1 can do W1 work. In D1 days working H1 hr/day for Rs. R1 and M2 man can do W2 work in D2 days working H2 hr/day for Rs. R2 Then,
The formula is given as:
( { M1 times D1 times H1 over w1 times R1 } = { M2 times D2 times H2 over W2 times R2 } )
Where,
M = no. of person
D = no. Of days
H = no. Of hours
W = work
R = rate(in Rs.)
Example 5: If 10 men complete half work in 12 days when they work 8 hours per day, In how many days 18 men complete the full work when they work 6 hours over day?
Solution: By formula
( { M1 times D1 times H1 over w1 } = { M2 times D2 times H2 over W2 } )
( {10 times 12 times 8 over {1 over 2}} ) ( = {18 times D2 times 6 over 1} )
(
10 times 12 times 8 times 2 = 18 times D2 times 6 )
( D2 = {160 over 9} ) ( = 17{7 over 9} ) Days (Ans)
If a men or b women can do a piece of work in d days then x men and y women together finish the whole work :
( D = {a times b times d over xb + ya} )
If men or b women or c child women can do a piece of work in d days then x men, y women, and z children together finish the whole work :
( {D = a times b times c times d over xbc + yac + zab} )
Example 6 : 3 men or 4 women can do a piece of work in 43 Days. In how many days 7 men & 5 women can do The same work?
Solution: 3m or 4w = 43 days
7m + 5 w = ?
From formula :
M1 D1 = M2 D2
( 3 m times 43 = (7m + 5w) times D2 )
( 129 m = {7m + 15 over 4 m} times D2 )
( 129 m = {43m over 4} times D2 )
D2 = 12 days ( answer)
Short trick :
( D = {a times b times d over xb + ya} )
( D = {3 times 4 times 43 over 28 + 15} ) = 12 days (answer)
Example 7: 1 man or 2 boys or 3 girls can do a piece of work In 88 days . In how many days one man , one boy And one girl can do the same work ?
Solution: 1m = 2b = 3g = 88 days
(1m + 1b + 1g)= ?
From formula :
M1 D1 = M2 D2
( 3g times 88 = (1m + 1b + 1g) times D2 )
( 3g times 88 = {3g + 3 over 2g + 1g}D2 )
( 3g times 88 = {6g + 3g + 2g over 2}D2 )
( 3g times 88 = {11 over 2 g times D2} )
D2 = 48 days (answer)
Short trick :
( D = {a times b times c times d over xbc + yac + zab )
( D =
{1 times 2 times 3 times 88 over 6 + 2 + 3} )
48 days (answer)
