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This time we are going to solve work and time related shortcut tricks. we are also aware that this chapter is important for competitive exams from Time and work sections objective question are asked in various exam like IBPS/SBI, Bank PO, SSC, UPSC, etc. Time and Work sections are time consuming chapter.now, we are going to solve work and time illustrations with related important points.

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Time and work aptitude questions are asked in every competitive exam. Placement papers for TCS, Infosys, Wipro, CTS, HCL, IBM or Bank exam or MBA exams like CAT, XAT, MAT, or other exams like GRE, GMAT tests always contain one or more aptitude questions from this section of quantitative aptitude. Problems on time and work which appear in CAT exams are quite advanced and complicated – But they can be solved easily if you know the basic formulas, shortcuts and tricks.

If know about these important points and short methods you can solve these question in seconds.Certain general important points are given underneath. the competitors will learn them truly valuable while taking care of issues identified with Time and Work.:

Important Terms for Time and Work Formulas-

Before we plunge into time and work formulas, let’s quickly go through the different terms that make up time and work formulas-

Work (W) – The total amount effort or labour required to complete a given task is called work. It is represented by the letter ‘W’ and is always considered as a whole or one in every problem.

Person/ People (P) – The number of individuals involved in completing the given work is called people/person. It is represented by the letter ‘P’.

Time (T) – The exact count taken by the clock to complete a certain task is called time. It is represented by the letter ‘T’. Time can be calculated in two ways- the number or days ‘D’ or the number of hours ‘H’.

Rate of Work (R) – The amount of work done in a unit of time is called the rate of work. It can be calculated on a daily or hourly basis, depending on the conditions given in the questions. It is represented by the letter ‘R’.

Important Point Related To Time & Work-

1. While solving problems related to time and work we have to compare the work of several working agents (men,women and children). to solve this purpose, we need find out the amount of work done by each agent in a common unit time. this unit time is generally taken as an hour, or minute of the problem.

2. The total work is assumed to be 1.

3. If A can do a piece of work in n days, then A’s 1 day’s work = 1/n.

4. If one can do a piece of work in 12 days, then it means that he can do 1/12 of work in 1 day.

5. Conversely, if one can do 1/12 work in 1 day, it implies that he can complete the work in 12 days.

6. One day’s work = 1/No. of days Required to finish the work

7. No. of days required to finish the work = 1/one day’s work

8. If A does 1/15 of work in 1 day and B does 1/10 of the same work in 1 day then A + B together will do 1/15 + 1/10 i.e. 1/6 of that work in one day.

9. If A is thrice as good as a work man as B then :

A’s Work : B’s Work = 3 : 1

And ratio of the times taken by A and B to complete the work = 1 : 3 i.e. A will take 1/3 of the time taken by B to complete that work.

10. The time required to complete a certain peice of work is increased or decreased in the same ratio in which the number of men engaged to do a peice of work be changed in the ratio 4 : 5, the time required to complete that peice of work will change in the ratio 5 : 4.

11. Time and number of man each are always in dirrectly proportional to work.

All the above points can be summarised as given below:

if ‘M1’ persons can do ‘W1’ works in ‘D1’ days and ‘M2’ persons can do ‘w2’ works in ‘D2’ days then we have a very general formula in the relationship of:

M1D1W2 = M2D2W1

it is all in one formula and basic formula to solve these type questions.

we can also include that:

(a) more men less days and conversely more days less men.

(b) more men more work and conversely more work more men.

12. If ‘A’ can do a peice of work in X days and ‘B’ can do it in Y days then ‘A’ and ‘B’ working together can do the same work in:

XY/X+Y days.

13. If A, B and c can do a work in X, Y and Z days respectively then all of them working together can finish the work in:

XYZ/XY+YZ+XZ days.

Shortcuts for frequently asked time and work problems

A and B can do a piece of work in ′a′ days and ′b′ days respectively, then working together:
1. They will complete the work in $\frac{a b}{a + b}$ days
2. In one day, they will finish ${(\frac{a + b}{a b})}^{t h}$ part of work.
If $A$ can do a piece of work in $a$ days, $B$ can do in $b$ days and $C$ can do in $c$ days then,
$A, B and C together can finish the same work in \frac{a b c}{a b + b c + c a} days$

$A, B and C together can finish the same work in \frac{a b c}{a b + b c + c a} days$
If $A$ can do a work in $x$ days and $A$ and $B$ together can do the same work in $y$ days then, Number of days required to complete the work if B works
$Number of days required to complete the work if B works alone = \frac{x y}{x - y} d a y s$ the work if B works alone=xyx-ydays

If A and B together can do a piece of work in x days, B and C together can do it in y days and C and A together can do it in z days, then number of days required to do the same work:
1. If A, B, and C working together = $\frac{2 x y z}{x y + y z + z x}$
2. If A working alone = $\frac{2 x y z}{x y + y z - z x}$
3. If B working alone = $\frac{2 x y z}{- x y + y z + z x}$
4. If C working alone = $\frac{2 x y z}{x y - y z + z x}$
If $A$ and $B$ can together complete a job in $x$ days.
If $A$ alone does the work and takes $a$ days more than $A$ and $B$ working together.
If $B$ alone does the work and takes $b$ days more than $A$ and $B$ working together.

Then, $x = \sqrt{a b} days$
If $m_{1}$ men or $b_{1}$ boys can complete a work in $D$ days, then $m_{2}$ men and $b_{2}$ boys can complete the same work in $\frac{D m_{1} b_{1}}{m_{2} b_{1} + m_{1} b_{2}}$ days.
If $m$ men or $w$ women or $b$ boys can do work in $D$ days, then 1 man, 1 woman and 1 boy together can together do the same work in $\frac{D m w b}{m w + w b + b m}$ days
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 inverse ratio. ie; $b : a$
If people work for same number of days, ratio in which the total money earned has to be shared is the ratio of work done per day by each one of them.
$A$ , $B$ , $C$ can do a piece of work in $x$ , $y$ , $z$ days respectively. The ratio in which the amount earned should be shared is $\frac{1}{x} : \frac{1}{y} : \frac{1}{z} = y z : z x : x y$
If people work for different number of days, ratio in which the total money earned has to be shared is the ratio of work done by each one of them.

Special cases of time and work problems

Given a number of people work together/alone for different time periods to complete a work, for eg: $A$ and $B$ work together for few days, then $C$ joins them, after few days $B$ leaves the job. To solve such problems, following procedure can be adopted.
1. Let the entire job be completed in $D$ days.
2. Let sum of parts of the work completed by each person = 1.
3. Find out part of work done by each person with respect to $D$ . This can be easily found out if you calculate how many days each person worked with respect to $D$ .
4. Substitute values found out in Step 3 in Step 2 and solve the equation to get unknowns.
A certain no of men can do the work in $D$ days. If there were $m$ more men, the work can be done in $d$ days less. How many men were there initially?
Let the initial number of men be $M$
Number of man days to complete work = $M D$
If there are $M + m$ men, days taken = $D - d$
So, man days = $(M + m) (D - d)$
ie; $M D = (M + m) (D - d)$
$M (D - (D - d)) = m (D - d)$

$M = \frac{m (D - d)}{d}$
A certain no of men can do the work in $D$ days. If there were $m$ less men, the work can be done in $d$ days more. How many men were initially?
Let the initial number of men be $M$
Number of man days to complete work = $M D$
If there are $M - m$ men, days taken = $D + d$
So, man days = $(M - m) (D + d)$
ie; $M D = (M - m) (D + d)$
$M (D + d - D) = m (D + d)$

$M = \frac{m (D + d)}{d}$
Given $A$ takes $a$ days to do work. $B$ takes $b$ days to do the same work. Now $A$ and $B$ started the work together and $n$ days before the completion of work $A$ leaves the job. Find the total number of days taken to complete work?
Let $D$ be the total number of days to complete work.
$A$ and $B$ work together for $D - n$ days.
So, $(D - n) (\frac{1}{a} + \frac{1}{b}) + n (\frac{1}{b}) = 1$
$D (\frac{1}{a} + \frac{1}{b}) - \frac{n}{a} - \frac{n}{b} + \frac{n}{b} = 1$
$D (\frac{1}{a} + \frac{1}{b}) = \frac{n + a}{a}$

$D = \frac{b (n + a)}{a + b}$ days.

Illustrations Related To Time & Work-

Illustration 1: A can do a peice of work in 4 days 5 hours each and B can do it in 5 days 6 hours each. How long will they take to do it, working together 1/6 hours a day?

A can complete the work in 4 × 5 = 20 hours.

B can complete the work in 5 × 6 = 30 hours.

A’s 1 hours work = 1/20 and B’s 1 hour’s work = 1/30

(A + B)’s 1 hour’s work = (1/20 + 1/30) => 1/12

Both will finish the work in 12 hours

Number of days of 1/6 hrs. each = (12 × 1/6) => 2 days.

Illustration 2: 3 men and 5 boys can do a peice of work in 20 days while 5 men and 3 boys can do same work in 15 days. In how many days can 4 men and 2 boy do the work?

let 1 men’s 1 day’s work = X and 1 boy’s 1 day’s work = Y

Then, 3X + 5Y = 1/20

And 5X + 3Y = 1/15

Solving, We get :

X = 13/64

Y = 1/320

(4 men + 2 boy)’s 1 day’s work = (4 × 13/64 + 2 × 1/320) = 131/160

So, 4 men And 2 boy together can finish the work in 160/131 = 29 days

Dr. R. S. Aggarwal Objective Type Questions, Try Yourself-

Question 1: A, B and c can complete a piece of work in 24, 6 and 12 days respectively. working together, they will complete the same work in:

(a) 1/24 days (b) 7/24 days (c) 3 3/7 days (d) 4 days

Question 2: A man can do a piece of work in 5 days, but with the help of his son, he can do it in 3 days. In what time can the son do it alone

(a) 6 1/2 days (b) 7 days (c) 7 1/2 days (d) 8 days