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Common Information

Working together, A and B can do a job in 6 days. B and C can do the same job in 10 days, while C and A can do it in 7.5 days.

Q.

Common Information Question: 1/2

How many days will it take if all A, B and C work together to complete the job?

 A.

8

 B.

5

 C.

3

 D.

7

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Solution:
Option(B) is correct

Even before you start working on the problem, check out if you can eliminate some answer choices as impossible.

We know that if A and B alone work, they can complete the job in 6 days. Therefore, if all three of them A, B and C work together the number of days it will take to complete the job will surely be less than 6 days. Hence, we can eliminate answer choices (1) and (4) right away.

Let A be the number of days that A will take to complete the job alone, B days for B to complete the job alone and C days for C to complete the job alone.

A and B can do a job in 6 days. They complete \(\left(\dfrac{1}{6}\right)^{th}\) of the job in a day

\(\dfrac{1}{A}+\dfrac{1}{B}=\dfrac{1}{6}\)........(1)

Similarly, B and C will complete \(\left(\dfrac{1}{10}\right)^{th}\) of the job in a day.

\(\dfrac{1}{C}+\dfrac{1}{B}=\dfrac{1}{10}\).....(2)

And C and A will complete \(\dfrac{1}{7.5}\) or \(\left(\dfrac{2}{15}\right)^{th}\)of the job in a day

\(\dfrac{1}{C}+\dfrac{1}{A}=\dfrac{2}{15}\)......(3)

Adding (1), (2) and (3) we get:

\(\dfrac{2}{A}+\dfrac{2}{B}+\dfrac{2}{C}=\dfrac{5+3+4}{30}\)

\(\dfrac{1}{A}+\dfrac{1}{B}+\dfrac{1}{C}=\dfrac{1}{5}\)

i.e working together, A, B and C complete \(\left(\dfrac{1}{5}\right)^{th}\) of the job in a day.

Therefore, they will complete the job in 5 days.

Edit: For an alternative solution, check comment by Aida.


(2) Comment(s)


Arvind
 ()

Arvind

AB BC. CA. Work

6 10 7.5 30(l.c.m)

5 3 4

2(A+B+C). = 12

A+B+C =6

=30/6. =5 days



Aida
 ()

$2(A+B+C)=\dfrac{1}{6} + \dfrac{1}{10} +\dfrac{1}{7.5}$

$2(A+B+C)=\dfrac{10}{60} +\dfrac{6}{60} +\dfrac{8}{60}=\dfrac{24}{60}$

$(A+B+C)=\dfrac{24}{60} : 2$ $= \dfrac{24}{120} \text{ in a day}$

Whole work $\dfrac{120}{24}= \textbf{5 days}$