A and B together can do a work in 8 days, B and C together in 6 days, while C and A together in 10 days. If they all work together, the work will be completed in how many days?a)250/47b)230/37c)240/47d)240/57Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Given:
A+B=1/8
B+C=1/6
C+A=1/10

To find:
If they all work together, the work will be completed in how many days?

Solution:
Let's assume that A, B, and C can complete the work in x, y, and z days respectively if they work alone.

A+B=1/8
=> (A+B)*8=1
=> 8A+8B=1
=> 4A+4B=1/2

B+C=1/6
=> (B+C)*6=1
=> 6B+6C=1
=> 3B+3C=1/2

C+A=1/10
=> (C+A)*10=1
=> 10C+10A=1
=> 5C+5A=1/2

Adding all three equations, we get:
4A+4B+3B+3C+5C+5A=3/2
=> 9A+9B+8C=3/2
=> 18A+18B+16C=3

Now, we need to find the value of A+B+C if they all work together.
Adding all three equations, we get:
2(A+B+C)=1/8+1/6+1/10
=> 2(A+B+C)=47/240
=> A+B+C=47/480

Let's assume that they all work together for 'd' days to complete the work.
So, the amount of work done by A in 'd' days = d/x
Similarly, the amount of work done by B in 'd' days = d/y
And, the amount of work done by C in 'd' days = d/z

As they all work together:
d/x + d/y + d/z = 1

Now, we need to find the value of 'd':
d = (x*y*z)/(xy+yz+zx)

Substituting the values of x, y, and z, we get:
d = (48*30*20)/(240*30+47*20+48*47)
=> d=240/47

Therefore, if they all work together, the work will be completed in 240/47 days (approx. 5.10 days).

Hence, option (c) is the correct answer.

Answer

Formula: A and B together can do a work in x days, B and C together in y days, while C and A together in z days. If they all work together, the work will be completed in: (2xyz)/( xy+yz+zx)

In the above given problem 'x' = 8 days, 'y' = 6 days and 'z' = 10 days.

Substituting the values x, y and z in the formula, If they all work together, the work will be completed in : (2 * 8 * 6 * 10)/((8*6)+(6*10)+(10*8))

⇒ 960/(48+60+80) = 960/188 = 240/47 days

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