Time and work Important Formulas

Time and work - Aptitude Important Formulas, Shortcut Methods and Tricks:

1. If ‘A’ can do a piece of work in ‘a’ days, then work performed in one day = 1/a part of work.

2. If A’s 1 day's work = (1/a), then ‘A’ can finish the work in ‘a’ days.

3. A is thrice as good a workman as B, then:
             Ratio of work done by A and B = 3 : 1.
             Ratio of times taken by A and B to finish a work = 1 : 3.

4. If ‘A’ is ‘x’ times as good as work performed as B, then time taken by A = (1/x)th time taken by ‘B’.

5. If ‘A’ and ‘B’ can do a piece of work in ‘x’ and ‘y’ days respectively, then time taken with working together = xy/(x + y) days.

6. If the number of men to do a job is changed in the ratio a : b, then the time required to do the work will be changed in the ratio b : a.

7. If two men ‘A’ and ‘B’ together can finish a job in ‘x’ days and if ‘A’ working alone takes ‘a’ days more than “A and B” working together and ‘B’ working alone takes ‘b’ days more than “A and B” working together,
             then, x = √ab



Example: 01
A and B can do a work in 10 days,  B and C can do the same work in 20 days, while C and A can do it in 15 days. In how many days can C alone do the same work?

Solution:
(A + B)’s 1 day’s work = 1/10
(B + C)’s 1 day’s work = 1/20
(C + A)’s 1 day’s work = 1/15
On adding,
2 (A + B + C)’s 1 day’s work = [1/10 + 1/20 + 1/15] = [(6 + 3 + 4)/60] = 13/60
(A + B + C)’s 1 day’s work = 13/120
Now, C’s 1 day’s work = {(A + B + C)’s 1 day’s work} - {(A + B)’s 1 day’s work}
          = (13/120) - (1/10)
          = (13 - 12)/120
          = 1/120
Hence, C alone can do the work in 120 days.

Example: 02
A and B could do a piece of work in 30 days. After working for 10 days, they are assisted by C and the work is finished in 10 days. If C does as much work in 2 days as B does in 3 days, in how many days could A do the same work alone?

Solution:
(A + B)’s 1 day’s work = 1/30 ………………………….(i)
(A + B)’s 10 day’s work = 10 × (1/30) = 1/3
Remaining work = 1 - (1/3) = 2/3
Now, since (A + B + C)’s 10 day’s work = 2/3
(A + B + C)’s 1 day’s work = 2/(3 × 10) = 1/15 ……………(ii)
From equation (ii) - (i), C’s 1 day’s work = (1/15) - (1/30) = 1/30
Hence, C can finish the work in 30 days.
Now, it is given that C does as much work in 2 days as B does in 3 days.
∴ The work which C does in 30 days, will be done by B in = 30 × (3/2) = 45 days.
Hence, B alone can finish the work in 45 days.
∴ B’s 1 day’s work = 1/45 ……………………(iii)
From equation (i) - (iii), A’s 1 day’s work = (1/30) - (1/45) = 1/90
Therefore, A alone can finish the work in 90 days.
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