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)