## Quantitative Aptitude Notes: Time and Work (Part - 1)

### Quantitative Aptitude Notes: Time and Work (Part - 1)

What is Work?
Technically Work is the quantity of energy transferred from one system to another but for questions based on this topic, Work is defined as the amount of job assigned or the amount of job actually done.
Analogy with Time – Speed – Distance:
Problem on work are based on the application of concept of ratio of time and speed. Work is always considered as a whole or one. There exists an analogy between the time-speed-distance problems and work.
Above mentioned definition of work throws light on three important points.
a) Work = 1 (It is always measured as a whole) = Distance
b) Rate at which work is done = speed
c) Number of days required to do the work = Time
Important Formulae:
Work from Days:
If A can do a piece of work in n days, then A's 1 day's work = 1/n
Days from Work:
If A's 1 day's work =1/n, then A can finish the work in n days
Ratio:
1. If A is thrice as good a workman as B, then:
a) Ratio of work done by A and B to finish the work = 3 : 1
b) Ratio of times taken by A and B to finish the work = 1 : 3
2. If A is x times as good as a workman B, then he will take (1/x)th of the time by B to do the same work.
3. A and B can do a piece of work in ‘a’ days and ‘b’ days respectively, then working together they will take [xy/(x + y)] days to finish the work and in one day, they will finish [(x + y)/xy]th part of the work.
4. If P1 persons working H1 hours a day can complete W1 units of work in D1 Days and P2 persons working H2 hours a day can complete W2 units of work in D2 Days then
P1H1D1/W1 = P2H2D2/W2
5. If three persons A, B and C can finish a work separately in t1, t2, t3 days respectively and work is finished after A works is finished after A works on it for d1 days, B works for d2 days and C works for d3 days then
[(d1/t1) + (d2/t2) + (d3/t3)] = 1

Memory Based Example Problems based on various types
Type 1 – Based on Two persons
1. A sum of money is sufficient to pay P’s wages for 25 days or Q’s wages for 30 days. The money is sufficient to pay the wages of both for
Solution:
If A can complete a work in x days and B can complete the same work in y days, then, both of them together can complete the work in x y/ x+ y days.
Here, wages can be taken as days,
Then, 25 ×30/ 55 = 13 7/11 days
2. Ravi and Kumar are working on an assignment. Ravi takes 6 hours to type 32 pages on a computer, while Kumar takes 5 hours to type 40 pages. How much time will they take, working together on two different computers to type an assignment of 110 pages?
Solution:
Number of pages typed by Ravi in 1 hour = 32/6 = 16/3
Number of pages typed by Kumar in 1 hour = 40/5 = 8
Number of pages typed by both in 1 hour = (16/3 + 8) = 40/3
Therefore, time taken by both to type 110 pages = (110 × 3/40) hours = 8 ¼ hours or 8 hours 15 minutes
3. Xavier can do a job in 40 days. He worked on it for 5 days and then Paes finished it in 21 days. In how many days Xavier and Paes can finish the work?
Solution:
Xavier’s one day’s work = 1/40
His 5 days work = 1/40 × 5 = 1/8
The work remaining = 1 – 1/8 = 7/8
7/8 of the work is done by Paes in 21 days
Then, his 1 day’s work = (7/8)/21 = 7/(21 × 8) = 1/24
Therefore, the time taken by Paes to finish the whole work = 24 days.
Then, The No. of days two of them together take to finish the work = 40 × 24/(40 + 24) = 40 × 24/64 = 15 days.
4. A alone can complete a work in 5 days more than A+B together and B alone can complete a work in 45 days more than A+B together. Then in how many days A and B together can complete the work?
Solution:
Shortcut = √(5×45) = 15
OR
Let (A+B) can do in x days, so
1/(x+5) + 1/(x+45) = 1/x
Solve, x2 = 225, x = 15
Type 2 – Based on Three persons (Individual data)
1. A can do a particular work in 6 days. B can do the same work in 8 days. A and B signed to do it for Rs. 3200. They completed the work in 3 days with the help of C. How much is to be paid to C?
Solution:
Amount of work A can do in 1 day = 1/6
Amount of work B can do in 1 day = 1/8
Amount of work A + B can do in 1 day = 1/6 + 1/8 = 7/24
Amount of work A + B + C can do = 1/3
Amount of work C can do in 1 day = 1/3 - 7/24 = 1/24
Work A can do in 1 day: work B can do in 1 day: work C can do in 1 day = 1/6 : 1/8 : 1/24 = 4 : 3 : 1
Amount to be paid to C = 3200 × (1/8) = 400
2. Machine P can print one lakh books in 8 hours. Machine Q can print the same number of books in 10 hours while machine R can print the same in 12 hours. All the machines started printing at 9 A.M. Machine P is stopped at 11 A.M. and the remaining two machines complete work. Approximately at what time will the printing of one lakh books be completed?
Solution:
Work done by P in 1 hour = 1/8
Work done by Q in 1 hour = 1/10
Work done by R in 1 hour = 1/12
Work done by P,Q and R in 1 hour = 1/8 + 1/10 + 1/12 = 37/120
Work done by Q and R in 1 hour = 1/10 + 1/12 = 22/120 = 11/6
From 9 am to 11 am, all the machines were operating.
i.e, they all operated for 2 hours and work completed = 2 × (37/120) = 37/60
Pending work = 1- 37/60 = 23/60
Hours taken by Q an R to complete the pending work = (23/60) / (11/60) = 23/11 which is approximately equal to 2
Hence the work will be completed approximately 2 hours after 11 am ; i.e around 1 pm
3. Chandra, Venky and Himu take Rs. 535 for doing a work together. If each takes 5, 6 and 7 days respectively to complete the work when working alone, what will be the share of Chandra?
Solution:
The share of the three will be in the share of inverse of their number of days of completion of work, that is in the ratio of 1/5: 1/6: 1/7.
Multiplying each of the member of the ratio by 210, the LCM of 5, 6, and 7, we get the ratio of the share as, 42 : 35 : 30. Let as assume the ratio to be, 42x: 35x: 30x
So the total share = 107x = Rs.535 => x = Rs.5
Therefore Chandra’s share 42x = Rs.210
Type 3 – Based on Three persons (Combined data)
1. P and Q can do a work in 30 days. Q and R can do the same work in 24 days and R and P in 20 days. They started the work together, but Q and R left after 10 days. How many days more will P take to finish the work?
Solution:
Let work done by P in 1 day = p, Work done by Q in 1 day = q, Work done by R in 1 day = r
p + q = 1/30; q + r = 1/24; r + p = 1/20
Adding all the above, 2p + 2q + 2r = 1/30 + 1/24+ 1/20 = 15/120 = 1/8 => p + q + r = 1/16
=> Work done by P,Q and R in 1 day = 1/16
Work done by P, Q and R in 10 days = 10 × (1/16) = 10/16 = 5/8
Remaining work = 1 = 5/8 = 3/8
Work done by P in 1 day = Work done by P,Q and R in 1 day - Work done by Q and R in 1 day = 1/16 – 1/24 = 1/48
Number of days P needs to work to complete the remaining work = (3/8) / (1/48) = 18
2. A and B together can do a piece of work in 12 days, which B and C together can do in 16 days. After A has been working at it for 5 days and B for 7 days, C finishes it in 13 days. In how many days C alone will do the work?
Solution:
According to the question,
A + B’s 1 day’s work = 1/12
B+ C’s 1 days’ work = 1/16
A worked for 5 days, B for 7 days and C for 13 days. So, we can assume that,
A+ B has been working for 5 days and B+ C has been working for 2 days and C alone for 11 days.
i.e. A’s 5 day’s work + B’s 7 day’s work + C’s 13 day’s work = 1
(A+ B)’s 5 days work + (B + C)’s 2 days work + C’s 11 day’s work = 1
5/12 + 2/16 + C’s 11 days work = 1
So, C’s 11 day’s work = 1 – (5/12 + 2/16) = 11/24
C’s 1 days’ work = 11/24/11 = 1/24
Therefore, C alone can finish the work in 24 days.
3. A and B can do a piece of work in 12 days; B and C can do it in 15 days and C and A can do it in 20 days. A alone can do the work in:
Solution:
2 (A + B + c)’s 1 day’s work = 1/12 + 1/15 + 1/20 = 12/60 = 1/5
A + B + C’s 1 day’s work = 1/5/2 = 1/10
A’s 1 day’s work = 1/10 – (B + C)’s 1 day’s work = 1/10 – 1/15 = 1/30
Therefore, A alone can finish the work in 30 days.