Let "a" be the rate of work of A, i.e. the part of work which A makes per day.

Let "b" be the rate of work of B, i.e. the part of work which B makes per day.


First statement of the problem says that the combined rate of work of A and B, working together,

is the value reciprocal to  8 = 

    a + b = .            (1)


The second statement of the problem says that

    8a + 10b = 1.           (2)



    Thus the setup is completed, and you have now two equations for two unknowns "a"  and  "b".



To solve this system of equations, multiply equation (1) by 8 (both sides).  Keep equation (2) as is.
You will get then

    8a +  8b =           (3)

    8a + 10b = 1             (4)


Next, subtract equation (3) from equation (4) (both sides).   You will get

          10b - 2b = 1 - ,   or

           2b      =  = .


Hence,  b =  = .

It means that B makes    of the job per day;  hence, B needs 146 days to complete the job alone.


Part of the problem is thus solved, and we need to find "a" now.


For it, from equation (2) you have

    8a = 1 - 10b = 1 -  = ;

hence,  a =  : 8 = .


It means that A needs   = 8   days to complete the job alone.