Question

5. Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in 3, Anne
and Carl in 5. How many days does it take all of them working together if Carl gets
injured at the end of the first day and can't come back?​

Answer 1

Given : Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in 3, Anne  and Carl in 5

To Find : How many days does it take all of them working together if Carl gets  injured at the end of the first day and can't come back

Solution:

Carl  Can complete work    = C

Bob  Can complete work   = B

Anne Can complete work    = A

Carl and Bob can demolish a building in 6 days,

=> 1/B  + 1/C = 1/6

Anne and Bob can do it in 3

=>  1/A + 1/B = 1/3

Anne  and Carl in 5 days

=> 1/A  +  1/C = 1/5

Adding all  

2(1/A + 1/B + 1/C )  = 1/6 + 1/3 + 1/5

=> 2(1/A + 1/B + 1/C )   = (5 + 10 + 6)/30

=>  1/A + 1/B + 1/C    = 21 /60

1/A + 1/B + 1/C -  (1/A + 1/B) =21 /60 -  1/3

=> 1/C = 1/60

Let say work completed in D Days   as carl works for 1 day only

then D/A + D/B + 1/C  = 1

=> D(1/A + 1/B) + 1/C = 1

=> D(1/3) + 1/60 = 1

=> D(1/3)  = 59/60

=>  D = 59/20

59/20 days = 2.95  days

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Answer 2

Given :

• Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in 3, Anne and Carl in 5

To Find :

• How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back ?

Solution :

$\large\blue{\boxed{\red{\sf Let}}}$

• Carl do the job in "c" days
• Anne do the job in "a" days
• Bob do the job in "b" days

$\large\blue{\boxed{\red{\sf Then}}}$

• The portion of job they can do is 1/c, 1/a and 1/b

As per given we have below equations:

• 1/c + 1/b = 1/6
• 1/a + 1/b = 1/3
• 1/a + 1/c = 1/5

Sum of all 3 equations given us:

$\implies$2(1/a + 1/b + 1/c) = 1/6 + 1/3 + 1/5

$\implies$2(1/a + 1/b + 1/c) = (5 + 10 + 6)/30

$\implies$2(1/a + 1/b + 1/c) = (21)/30

$\implies$2(1/a + 1/b + 1/c) = 7/10

$\implies$1/a + 1/b + 1/c = 7/20

It means all there together can complete 7/20 of the job in one day.

The rest of the job is done by Anne and Bob:

$\implies$1 - (7)/20 = (13)/20

As Anne and Bob can do 1/3 of the job in one day they need time to complete the rest:

$\implies$(13)/20 : 1/3 = (39)/20

Add one day to this to find overall time:

$\implies$1 + (39)/20 = (59)/20 days or 2.95 days