John can build a wall in 30 days and Peter can demolish the same wall

IMO E

LCM of 30 and 40 = 120

Rate of John = 4 unit /day

Rate of Peter = 3 unit/day (demolish will be -)
D1---D2----D3-----D4
4----(-3)-----4----(-3)

Work done will be 1 unit in 2 days
120 units will be done in 240 days

rate of John = 1/30
rate of Peter = 1/40
combined rate = 1/30-1/40 = 1/120 ; 2 days work

John rate for rest of the days would be (29/30/ 1/120 ) * 2 = 232
since John has been working since day 1 ; 232+ 1; 233 days

As question asks, we need to find the number of days to build the wall first. This implies that we can not let the Peter demolish wall anymore after the wall is built. Hence, John will work for x+1 days and peter will work for x days.

Let the wall represent 120 units of work.
John's rate is 4 units per day
Peter's is -3 units per day.

Combining the rates we get, 1 unit out of 120 units is completed in 2 days.

in x+1 days john can do 4(x+1) units of total work.

in x days peter can do 3x units of work. (negative work)

total work - combined work of peter and john = 0

let's put values in the equation,
120-(4(x+1)-3x)=0
120-x-4=0
x=116.... (no of days taken by peter)

x+1=117.... (no of days john took to build)

hence, 116+117=233...total no days taken by john and peter to end up with a built wall.

Posted from my mobile device

Work done by John in 1 day = \(\frac{1}{30}\)

Work done by Peter in 1 day = \(\frac{-1}{40}\)

Work done by them in 2 days is \(\frac{1}{30} - \frac{1}{40}\) = \(\frac{1}{120}\)

Work done in 1 day is \(\frac{1}{240}\)

So it takes 240 days to build the wall for first time.

OPTION: E

Hi VeritasKarishma,
Why do we have to assume that john will finish the work? Why can't peter finish the work?

Below is my analysis, please can you confirm if the same is correct or not?

I guess since Peter Demolishes then John would actually have to build a little bit more than the whole wall , for Peter to finish the wall.

So if Peter were to finish then , we would actually have built more than the whole wall, and the time taken would be 240 days.
But we are asked " When was the whole wall built for the first time" we have to let John Complete the wall.
So in the last day John has to do 1/30 of the work.
Hence we have to find how much time Both John and Peter will take to do 29/30 of the work, leaving 1/30 work for John to finish up.


Hence total days is time taken by John and Peter to do 29/30 of the work plus one day that John will take to finish.

Is this analysis correct?

Let the work be 120 units

John does 4 units per day and Peter does -3 units per day

Every two days = 1 unit added

232 days = 116 units added

233 days = 120 units added

Hi,

Work rate of John: 1/30
WR of Peter: -1/40
Days John works: x+1
Days Peter Works: x
John always works one more day because a) he starts, and b) the work will FIRST be done after John finishes it, and NEVER when Peter was the last one to work on it. This is because Peter always destroys parts of it, so it must first be finished after Peter.

-> 1=(1/30)(x+1)-(1/40)(x)
-> x=116, x+1=117

Adding both you get (B)

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