Time & Work

Time & Work 
    Time and Work problems  refers  that the amount of work done by an individual or
    a group of individuals in one day.

 Points are to be remember on solving questions on time and work 

    (1) The work to be completed is considered as one unit.

    (2) It is assumed that a person does the same amount of work each day.

    (3) Work and men are directly proportional. 

    (4) work and days are directly proportional. 

    (5) Men and days are inversely proportional.
Concept for solving problems related to Time and Work

Case I: If a person completes a piece of work in n days, then work done by person in one day = (1/n)^th  part of the work.

Case II: If a person  completes (1/n)^th  part of work in one day, then person will take n days to complete the work.

Example 1 :-If Ram and Shyam  can do a job in 10 days and 18 days independently,

                  how many days would they take to complete the same job working simultaneously?

Explanation :-Ram  can do a job in 10 days

                       Hence Ram one day work = 1/10

                       Shyam  can do a job in 18 days

                       Hence Shyam one day work = 1/18

                       Ram  and Shyam one day work together =1/10 + 1/18

                                                                                        = 14/90 = 7/45

                       Hence Ram  and Shyam can do the complete work 45/7 days .

Example 2 :-Raj can finish a work in 10 days and Ram can finish

                     the same work in 40 days. If Raj and Ram both work together

                     then what is the total number of days taken?

Explanation :-Raj  can do a job in 10 days

                       Hence Raj one day work = 1/10

                       Ram  can do a job in 40 days

                       Hence Shyam one day work = 1/40

                       Raj  and Ram one day work together =1/10 + 1/40

                                                                                        = 5/40 = 1/8

                       Hence Raj  and Ram can do the complete work 8 days .

Example 3 :-Shivam, kamal, Yash can finish a piece of work in 10 days,

                      12 days and 15 days respectively. If all the three work at it together,

                      they are paid Rs.1500 for the whole work. What is the share of Shivam?

Explanation :-Shivam’s 1 day’s work = 1/10                       

                         kamal’s 1 day’s work = 1/12

                        Yash’s 1 day’s work = 1/15

                       Therefore, ratio of shares of Shivam, kamal, Yash = Ratio of their 1 day work.

                                                                         = 1/10 : 1/12 : 1/15

                                                                              = 6 : 5 : 4

                               Sum of ratio terms = 6 + 5 + 4 = 15

                              Shivam gets =(6/15)*1500 = Rs. 600

Example 4 :-If A & B together  can finish a piece of work in 10 days,A alone 
                      can do the same work in 15 days .How many days would B take
                      to complete the same  work.

Explanation :-A & B together  can finish a piece of work in 10 days
                         A & B together  one day work  = 1/10     
                         A  can finish a piece of work in 15 days
                         A  one day work = 1/15                 

                        B  one day work in =  1/10 - 1/15 = 1/30
                        B together  can finish a piece of work in 30 days

Example 5 :-If A & B together  can do a piece of work in 18 days,B & C together
                     can do a piece of  work in 24 days,A & C together  can do a piece of work
                     in 36 days. In how many days will A,B and C finish it working together
                     and separately.

Explanation :-A + B together  can finish a piece of work in 18 days

                         A + B together  one day work  = 1/18     

                         B + C together  can finish a piece of work in 24 days

                         B + C together  one day work  = 1/24

                         A + C together  can finish a piece of work in 36 days

                         A + C together  one day work  = 1/36    

                         Adding these three

                        2(A+B+C) together  one day work  = 1/18 +1/24 +1/36

                                                                                 = 9/72 = 1/8

                        (A+B+C) together  one day work = 1/16

                     Hence A,B and C finish  work together in 16 days

                     A  one day work =(A+B+C)   one day work- (B + C)   one day work

                                                 =/16-1/24 = 1/48    

                    A  can finish a piece of work in 48 days

                   similarly  B  can finish a piece of work in 28 and 4/5 days

                        and C  can finish a piece of work in 144 days

Example 6 :-If A & B working separately  do a piece of work in 9 days,and 12 days
                     If they work for a day alternately A beginning  in how many days work
                     will be completed.

Explanation :-A  one day work =1/9,B one day work = 1/12
                         A + B  two days work  = 1/9 + 1/12 = 7/36
                         (A + B) 10 days work  = 35/36
                         remaining work = 1-35/36 = 1/136
                         on eleventh day A turn 1/9 work in 1 day
                        hence 1/36 work done by A in 1/4 day
                        Hence the work will be completed in 10 and 1/4 days.

Example 7 :- Ten men can complete a  piece of work in 15 days and fifteen women can
                       complete a  same work in 12 days ,if all 10 men and 15 women work
                       together ,in how many days work will be completed.
Explanation : Ten men  one day work =1/15
                         Fifteen women  one day work =1/12
                        (Ten men +Fifteen women)  one day work = 1/15+1/12 = 9/60
                                                                                            = 3/20
                       Hence 10 men and 15 women work
                       together  will complete the work in 20/3 days
Example 8 :- 45 men can complete a  piece of work in 16 days 6 days after they
                       started working ,30 more men joined them in how many days will they
                       now to take the complete remaining work..
Explanation : 45 men  one day work =1/16
                         one man  one day work =1/(16*45) = 1/720
                        45 men 6 days work = 6/16 = 3/8
                         Remaining work = 1-3/8 = 5/8
                        75 men one day work =75/720 = 5/48
                        Hence 5/48 work done by them in one day

                       Hence 5/8 work done by them (48/5)*(5/8) = 6 days