# Time and Work

In daily life, there are situations where we need to complete a particular job in a reasonable time. We have to complete the project earlier or later depending upon the needs. Accordingly, the men on duty have to be increased or decreased. The time and the men engaged for a project are inversely proportional to each other. The more the number of men involved, the lesser is the time required to finish a job.

### General Rules

1. If A can do a piece of work in n days, then A will finish 1/nth work in one day.

2. If A does three times faster work than B, then ratio of work done by A and B is 3:1 and ratio of time taken by A and B is 1:3.

3. 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/(X+Y) days.

A’s 1 day’s work = 1/X

B’s 1 day’s work = 1/Y

Then, (A + B)’s 1 day’s work = 1/X + 1/Y = (X+Y)/XY

A and B together can complete the work in XY/(X+Y) days.

4. There are two groups of people with same efficiency. In one M1 persons can do W1 works in D1 time and in the other M2 persons can do W2 works in D2 time. The relationship between the two groups is given by

(M1 × D1)/W1 = (M2 × D2)/W2

### Examples

Example 1: A can finish a piece of work by working alone in 6 days and B, while working alone, can finish the same work in 12 days. If both of them work together, then in how many days, the work will be finished?

Here, X = 6 and Y = 12

Working together, A and B will complete the work in (6 × 12)/(6 + 12) days.

= 4 days.

Example 2: A, B and C can complete a piece of work in 10, 15 and 18 days, respectively. In how many days would all of them complete the same work working together?

A’s 1 day’s work = 1/10

B’s 1 day’s work = 1/15

C’s 1 day’s work = 1/18

(A + B + C)’s 1 day’s work = 1/10 + 1/15 + 1/18

= 2/9

So, A, B and C together can complete the work in 9/2 or 4.5 days.

Example 3: A and B working together take 15 days to complete a piece of work. If A alone can do this work in 20 days, how long would B take to complete the same work?

A and B together can complete the work in 15 days.

(A + B)’s 1 day’s work = 1/15

Similarly, A’s 1 day’s work = 1/20

Therefore, B’s 1 day’s work = 1/15 - 1/20

= 1/60

B alone can complete the work in 60 days.

Example 4: A and B can do a piece of work in 12 days, B and C in 15 days, C and A in 20 days. How long would each take separately to do the same work?

(A + B)’s 1 day’s work = 1/12

(B + C)’s 1 day’s work = 1/15

(C + A)’s 1 day’s work = 1/20

So, [(A + B) + (B + C) + (C + A)]’s 1 day’s work = 1/12 + 1/15 + 1/20

2 (A + B + C)’s 1 day’s work = 1/5

(A + B + C)’s 1 day’s work = 1/10

A, B and C working together will complete the work in 10 days.

A’s 1 day’s work = (A + B + C)’s 1 day’s work – (B + C)’s 1 day’s work

= 1/10 - 1/15 = 1/30

A alone can do the work in 30 days.

Similarly, B's 1 day's work = 1/10 - 1/20 = 1/20

B alone can do the work in 20 days.

Similarly, C's 1 day's work = 1/10 - 1/12 = 1/60

C alone can do the work in 60 days.

Example 5: A and B together can do a piece of work in 3 days. If A does thrice as much work as B in a given time, find how long A alone would take to do the work?

Let A takes k days to do the work.

So, A's 1 day's work = 1/k

B's 1 day's work = 1/3k

(A + B)'s 1 day's work = 1/3 = 1/k + 1/3k

1/3 = 4/3k

k = 4

Example 6: A is thrice as good a workman as B and takes 10 days less to do a piece of work than B takes. Find the time in which B alone can complete the work?

Let B alone can complete the work in k days.

So, A takes (k - 10) days to complete the work.

A's 1 day's work = 3 (B's 1 day's work)

1/(k - 10) = 3/k

k = 3(k - 10)

k = 15

Example 7: If 10 persons can complete two-fifths of a work in 8 days, then find the number of persons required to complete the remaining work in 12 days?

M1 = 10, W1 = 2/5, D1 = 8

M2 = ?, W2 = 3/5, D2 = 12

(10 × 8)/(2/5) = (M2 × 12)/(3/5)

40 = 4M2

M2 = 10

Example 8: 12 men or 15 women can do a work in 14 days. In how many days, 7 men and 5 women would complete the work?

Work done by 1 man in 1 day = 1/(12 × 14)

Work done by 1 woman in 1 day = 1/(15 × 14)

(7 men + 5 women)'s 1 day's work = 7/(12 × 14) + 5/(15 × 14)

= 1/24 + 1/42 = 11/168

So, required number of days = 168/11 = 15.27 days