Time and Work
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Q.

A and B can do a piece of work in 45 days and 40 days respectively. They began to do the work together but A leaves after some days and then B completed the remaining work in 23 days.

The number of days after which A left the work was

 A.

12

 B.

11

 C.

10

 D.

9

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Solution:
Option(D) is correct

(A+B)'s 1 day's work = \(\dfrac{1}{45}+\dfrac{1}{40}=\dfrac{17}{360}\)

Work done by B in 23 days = \(1\times \dfrac{23}{40}=\dfrac{23}{40}\)

Remaining work = \(1-\dfrac{23}{40}=\dfrac{17}{40}\)

Now,  \(\dfrac{17}{360}\) work was done by (A+B) in 1 day.

\(\dfrac{17}{40}\) work was done by (A+B) in \(1\times \dfrac{360}{17}\times \dfrac{17}{40}=9\text{ days}\)

Therefore, A left after 9 days.

Edit: Thank you Abhishek Fauzdar for providing an alternate solution in the comments.

Edit 2: For yet another alternative solution, check comment by Doss.

Edit 3: For yet another alternative solution, check comment by Pasanna Jd.

Edit 4: For an alternative solution using Unitary Method, check comment by Chirag Goyal.

Edit 5: For an alternative solution making use of LCM method, check comment by Aishwarya.

Edit 6: For an alternative solution using % format, check comment by Sanajit Ghosh.

Edit 7: For yet another solution, check comment by Tushar.


(15) Comment(s)


Herat Patel
 ()

w/day

A 45 8 Let Total work = LCM=(45,40)=360

B 40 9

17

B alone do remaining work in 23 days

--> 23*9 = 207 work done by B

--> Remaining work (360-207) = 153

--> 153 work done by both A and B

--> 153/17

--> Ans : 9



Raj
 ()

Great effort on this website. Great going guys.



Tushar
 ()

A B

Time 45D 40D

Work per D 8 9

_______________________________

LCM 360

B completes remaining work in 23 days and B Rate of doing work is 9 Work per day = 23x9 = 207

Total Work - B's Work = 360-207 = 153 (Remaining Work)

Rate of A+B = 8+9 = 17

So A+B Doing their work till = 153/17 = 9 Days



Anurag
 ()

1 line answer:

x/45+(x+23)/40=1

solve it x=9



Sanajit Ghosh
 ()

For those who want to get in % format check the right way

A'S 1 DAY WORK= 2.22%

B'S 1 DAY WORK= 2.50%

A+B 'S 1 DAY WORK= 4.72%

B'S WORK FOR 23 DAYS = 23*2.5% =57.5%

REMAINING WORK =100-57.5%

= 42.5% (THIS WORK NEED TO BE COMPLETED BY BOTH )

so (2.20+2.50)*9=42.5% so 9 days



Aishwarya
 ()

Take $lcm = 360$ unit of work

For $A$,

$A= \dfrac{360}{45}=8$ units/day

For $B$,

$B= \dfrac{360}{40} = 9$ units/day

Job done by $B$ in 23 Days,

$9 \times 23=207$ units of work.

Work left,

$=360-207= 153 \text{ units}$

So, $A+B (=8+9=17)$ will finish the job in,

$=\dfrac{153}{17}$

$=\textbf{9 days}$


SRikanth
 ()

This method is help ful...

How to learn tims method to solve any other problems?

Any book available for learn this method?

Pls tel me


Chirag Goyal
 ()

Here's a $\text{Unitary Method}$

Let $\text{A & B}$ both Worked for $\text{X}$ Days together,

and $\text{B}$ Worked for Extra 23 days.

So $\text{(A + B's 1 day work)}\times{\text{X Days}}\text{ + (B's 1 day work)}\times{\text{23 Days}} = 1$

$(\dfrac{1}{45}\ + \dfrac{1}{40}){\text{X}} + (\dfrac{1}{40})\text{23} = 1$

$\text{X = 9 Days}$



Pasanna Jd
 ()

its simple if you go by question $A+B$'s 1 day work $= \dfrac{17}{360}$.

$A$ leaves after $X$ days hence both of them work for $X$ days ,work completed $= \dfrac{17 \times X}{360}$.-------- (1)

Now, $B$ does 23 days work alone to compelte the work,hence work done by $B$ in 23 days is $= \dfrac{23}{40}$ --------(2)

Hence, solving for $X \Rightarrow (1)+(2) = 1$ that gives $X=9$


Sudhanshu
 ()

Not much different from the original solution given, but a useful one, thank you :)


Naaz
 ()

thank you hamza for clarifying this.


Doss
 ()

This type of problem can be solved by using the shortcut.

Shortcut is:

No.of days it will take if $A$ and $B$ worked together * Fraction of days $B$ worked with $A$ (together).

Example: $A$ and $B$'s one day work is $\frac{1}{45}$ and $\frac{1}{40}$.

If they worked together they will complete the total work in $\dfrac{45*40}{(45+40)}$ days.

$B$ alone worked for 23 days out of $B$'s 40 days, so fraction will be $\dfrac{(40-23)}{40}$.

Finally $\dfrac{45*40*17}{(85*40)}=9$ days.

Summary:

In simple they are asking how many days both $A$ and $B$ are worked together.

So, we can find this using $A$ and $B$'s total days worked together $\dfrac{45*40}{(45+40)}$ and multiplying the fraction of days $B$ didn't work with $A$ i.e. multiply with $\dfrac{(40-23)}{40}$.

Thus, $\dfrac{45*40*17}{(85*40)}=9$ days is the final answer.


Abhishek Fauzdar
 ()

A'S 1 DAY WORK= 2.22%

B'S 1 DAY WORK= 2.50%

A+B 'S 1 DAY WORK= 4.72%

B'S WORK FOR 23 DAYS = 23*2.5% =47.5%

REMAINING WORK =100-47.5%

= 42.5% (THIS WORK NEED TO BE COMPLETED BY BOTH )

THEREFORE, => 42.5/4.72 = 9 DAYS


Naaz
 ()

$23 \times 2.5\%= 57.5\%$

Right?

How come its $47.5\%$

Please explain this step.

Hamza
 ()

You are right Naaz, but he has made another calculation mistake $100-47.5\%= 42.5\%$.

which should have been,

$100-57.5\%= 42.5\%$, making no difference in the final answer though.

So he messed up the calculations twice, correcting one mistake by another and finally ended up getting the correct ans.