Aptitude Discussion

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 |

**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.**

**Herat Patel**

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**Raj**

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Great effort on this website. Great going guys.

**Tushar**

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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**

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1 line answer:

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

solve it x=9

**Sanajit Ghosh**

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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**

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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,

$=\dfrac{153}{17}$

$=\textbf{9 days}$

**Chirag Goyal**

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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**

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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 $X=9$

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

**Naaz**

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thank you hamza for clarifying this.

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**

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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

$23 \times 2.5\%= 57.5\%$

Right?

How come its $47.5\%$

Please explain this step.

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.

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