Time And Work: Important Formulas And Solved Problems

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 :

1. More man can do more work
   
2. More work means more times required to do work
   
3. 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)