Leo, Shelly and Raj can complete a certain job in 10, 12, and 15 days

shouldn't the answer be A?
According to the question my soln comes 89/15. where am i going wrong?

Let's total unit of work LCM[10,12,15] = 60 units of work
Leo does 60 /10 = 6 units of work per day
shelly does 60/112 = 5 units of work per day
Raj does 60 / 15 = 4 units of work per day

now Raj leaves 2 before the completion day and leo leaves 1 day before the completion
Assume the work completed on x days
Therefore,
(6 + 4 +5)(x - 2) + (6 + 5) + 5= 60
==>15x - 30 = 60 - 16
==> 15x = 44 + 30 = 74

Hence x = 74 / 15

let d=days to complete job
d(15/60)-14/60=1
d=74/15 days
E

Responding to a pm:

On the last day of the job, only Shelly worked so she could have done only 1/12 of the work.

So 11/12 of the work was done before the completion day.

On the day before last, only Leo and Shelly worked so they would have done 1/10 + 1/12 = 11/60 of the work

So 11/12 - 11/60 = 11/15 of the work was done before by all three.
Combined rate of all three = 1/10 + 1/12 + 1/15 = 1/4
For this, time taken = (11/15) / (1/4) = 44/15

Total time taken = 44/15 + 2 = 74/15

Note: Usually in questions with fractional number of days, the fractional day is the last day. It is certainly unsettling that as per this question, the work was started in the middle of the day some time.

There's a typo in your first equation - it should be \(15(x-2) + 11 + 5\)

By the way, how do I understand that the completion day does not vary, i.e. it is fixed?

I've spent ~40 freaking mins trying to solve this, because I thought that the completion day changes once a person leaves the team, which is logical. That is, when Raj left, I changed the Work value to \(1/2\) and Rate value to \(11/60\). Using these values I calculated the time needed to finish the work - \(30/11\). Then I figured out that Leo leaves at moment \(30/11 - 11/11 = 19/11\). I calculated the amount of work done by both Leo and Shelly, which is \(11/60 * 19/11 = 19/60\).

Now, knowing that the amount of Work left is \(1 - (30/60) - (19/60) = (11/60)\), I calculated the time needed for Shelly to finish the work alone, which is \((11/60) : (5/60) = (11/5)\).

Finally, I added up all the time values, i.e. \(2 days + (19/11) days + (11/5) days\), and obtained a result \((326/55)\).

This did not match any of the answers provided here, and I tried to find my mistake for 40 FREAKING MINUTES. As you can see, my logic was not wrong. So, how should have I figured out what I was supposed to calculate?

Thanks for notifying Krotishka1 - Made the necessary change

Hey Krotishka1

The mistake you are making is that you have calculated the time for Leo and Shelly to complete
the work, but the work doesn't get completed. When you are adding up the individual times you
have added the time that Leo and Shelly would have taken to complete the work

Whenever you want to solve this kind of problem using this method you need to go backward from
the last day to the first day. The solution provided by VeritasPrepKarishma should make it clear.

Hope this helps you.

Is there a different way to solve this? The traditional way of taking what work would be done on a day to day basis.

Let the total work be 60 units....

So, Individual efficiencies are -

Leo : 6 units/day , Shelly : 5 units/day & Raj : 4 Units/day

Now, consider that the total work was considered in "t" days and from this part

We understand that only shelly worked for "t" days....

Now, \(6*(t - 1) + 4*(t - 2) + 5t = 60\)

Or, \(6t - 6 + 4t - 8 + 5t = 60\)

Or, \(15t - 14 = 60\)

Or, \(t = 74/15\), Answer must be (E)

Let work is completed in n days
Then
n/12 + (n-1)/10 + (n-2)/15 = 1
15n - 14 = 60
n = 74/15

Posted from my mobile device

Let work is completed in n days
Then
n/12 + (n-1)/10 + (n-2)/15 = 1
15n - 14 = 60
n = 74/15

Posted from my mobile device

­Rate of work for Shelly, Leo and Raj with W being 1 work unit -
1. Shelly = 1/12 W/day
2. Leo = 1/10 W/day
3. Raj = 1/15 W/day

Now solving these using the timeline method -

start |-------L+R+S-------|-------S+L-------|-------S-------|
                             [-2 days]         [ -1 days]

Assuming total number of days is X - 

Total time Shelly works is X x 1/12
Total time Leo works is (X - 1) x 1/10
Total time Raj works is (X - 2) x 1/15

All of this adds up to 1 Work unit -

X x 1/12 + (X-1) x 1/10 + (X-2) x 1/15 = 1

Solving for X we get => 74/15 days