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__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 P

_{1}persons working H

_{1}hours a day can complete W

_{1}units of work in D

_{1}Days and P

_{2}persons working H

_{2}hours a day can complete W

_{2}units of work in D

_{2}Days then

P

_{1}H_{1}D_{1}/W_{1}= P_{2}H_{2}D_{2}/W_{2}**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

[(d

_{1}/t_{1}) + (d_{2}/t_{2}) + (d_{3}/t_{3})] = 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.