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A and B working separately can do a work in 9 and 12 days respectively. If they work for a day alternately, after A start the work, the work will be completed  in ________ days.

work done by A in single day = $\frac{1}{9}$

work done by B in single day = $\frac{1}{12}$

now both works in alternate day, that means in 2 days total work can be done is $\frac{1}{9} + \frac{1}{12}$ _________________(1)

if both works in alternate day and work can be done in $x$ days than work done combinely in single day = $\frac{1}{x}$

so work done in 2 days = $\frac{2}{x}$ _______________________(2)

on comparing both $\frac{1}{9} + \frac{1}{12} = \frac{2}{x}$

$\therefore\text{ x = 10.28 days for combine work}$
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thanks
X=10.25 days precisely...here is how

‘A’ take 9 day for work. Lets say work is 1 unit.

We can say A has rate of work as 1/9 unit of work per day

‘B’ take 12 day for work.
rate of B = 1/12 work/day

To make denominators equal,

A=1/9 =4/36

B=1/12=3/36

now A starts the work and B follows and they do this alternately, so every 2 days they do

4/36 + 3/36= 7/36 work

we need total 1 unit of work means 36/36 of work.

so 7*5=35, since we take 7/36 as work of two days, 7*5/36 will be work of 10 days.

Now on the last day it will be A whose turn it is, and its speed is 4/36 , we only need 1/36 more to complete work on 11th day. So A will work only quarter of the day i.e 6 hours.

Thus answer comes out to be 10.25 days.

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