How to solve Time & Work (Shortcut Approach)

While attempting questions, you must remember some important points – 

1. Read the questions carefully.

2. Words like together, alone, before complete, after complete, etc. are important.

3. Data provided in the question should be read carefully.

How to approach and express data while taking the exam?

If there are two-person A and B. If A begins and both work alternate days. It means

1st day	2nd day	3rd day	4th day	5thday	6thday	7thday	8thday	9thday	10thday	…..
A	B	A	B	A	B	A	B	A	B	…..

 Before going forward, we will discuss some basics of Ratios to make calculation easy for this topic and these basics will also help you in other topics like Partnership, time and distance, average, allegation, etc.

TIME AND WORK EXAMPLES:

Example 5: A and B can do work in 8 days and 12 days respectively.

                     A                B              

Time (T)         8 days      12 days        

Efficiency (η)     12             8              

        η             3               2

total work  = T_A× η_A  = T_B× η_B
total work =  8 × 3 =  12 × 2 
total work = 24 units

(a). If both A and B work alternatively and  A begins, then in how many days work will be completed?

According to the question, A begins and both and b work alternately.

Days	1st day	2nd day	3rd day	4th day	5th day	6th day	7th day	8th day	9th day	10th /2-day total	Total Work
Person	A	B	A	B	A	B	A	B	A	B
Work Unit	3	2	3	2	3	2	3	2	3	1	24

Hence work will be completed in 9 (1/2) days.

But this type of approach is not helpful in exams. We will go by basic but in a smarter way.

If we make a pair of A and B, we can say that both together will work 5 units but in 2 days.

2 days = 5 units of work

2 days × 4 = 5 units of work × 4 

8 Days   = 20 units of work, i.e. we can say that up to the 8^th day 20 units of work will be done

But on the 9th day, it’s a chance of A and he will do his 3 units of work. So up to the 9^th day, 23 units of work will be done. Now 1 unit of work will remain. On the 10^th day, it’s a chance of B.

As above B does 2 units of work in a day. So, he will do 1 unit of work in (1/2) days.

So total time taken to complete the work is 9 (1/2) days.

(b). If A and B both work for 4 days then A leaves. In how many days total work will be completed.

(Eff_A+B× t_A+B )+( Eff._B × TB_B) = total work
(5 × 4)+(2 ×TB_B) = 24
20 + 2 × to_B  = 24
2 ×to_B = 4
t_B = 2 days , hence total time will be ( 4+2 )days i.e. 6 days.

Important: The same question can be framed in many ways.

Way 1: B works for 2 days and after that A joins B, then in how many days the work will be completed.

Solution: B works only for 2 days and then A joins B, i.e. both will work together after 2 days.
(B_B × η_B) + (t_A+B× η_A+B) = total work
(2 × 2) + (t_A+B × 5) = 24
(t_A+B × 5) = 20
t_A+B  = 4 days   hence total time is (2+4) days i.e. 6 days

Way 2: A and B both work together and A takes leaves for 2 days, then in how many days the work will be completed.

Solution:  When both A and B are working together and A takes leaves for two days it means B has to work alone for 2 days 

Let total time to complete the work is t days.

So,  η­_A+B × (t-2) + η_B × 2 days = 24
5 × (t-2) + 2 × 2 = 24
t-2 = 4
t = 6 days Hence total time taken to complete the work is 6 days.

Don’t confuse between t_A+B and T_A+B  (as mentioned in the article1). Both are different.

Example 6: X can do work in 6 days, Y can do it in 8 days and Z can do it in 12 days.

(a). If X starts the work and X, Y, Z works on alternate days, then in how many days the work will be completed?

Solution: Here X will start work on the 1^st  day, then Y will work on the 2^nd day, and z will work on the 3^rd day.        

                  X:   Y:  Z                         
Time           6 :   8  : 12                

Efficiency    12 × 8: 6 × 12 : 8 × 6              

η               96 :  72 : 48                           
η                 4 :  3  :  2                             

total work = η_X× T_X= η_Y×T_Y =η_Z×T_Z        

total work = 6 × 4 = 8 × 3 = 12 × 2 = 24 units

On 1^st day, work done by X = 4 unit

On 2^nd day, work done by Y = 3 unit

On 3^rd day, work done by Z = 2 unit

Total work in 3 days done by X,Y and Z = 9 unit

Now again X will come then Y, then Z and so on till work is completed.

                  In 3 days   =    9 units

                        × 2           × 2

                 In 6 days    =   18 units

X will work on 7^th day   = 4 units

                                  =  22 units

Now we need (24-22) units = 2 units work more but Y can do 3 units of work in one day. So 2 units will be done in (2/3) day.

               +(2/3) day          + 2 unit

Total days=7 (2/3) days       24 units

Hence total work will be done in 7 (2/3) days.

(b). If all started together and after completion of (3/4)^the work, Y left and remaining work is done by X and Z together. Then in how many days work will be completed?

Solution: Total work is 24 units then (3/4)^the work is 18 units.

One day work of (X+Y+Z) = 9 units so in 2 days 18 units of work will be done by (X+Y+Z) together.

After this Y left, X+Z worked together and 6 units of work remained.
One day work of X+Z = (4+2) units = 6 units. So in 3 days, total work will be completed.

Now, that you are clear with the concept, do attempt the quiz to make sure that you have understood it properly.

You can also go through the Part – I of Time and Work series from the link below –

How to solve Time & Work (Shortcut Approach) – Part I