Question 1163797
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If A and B work on a project, they need 8 1/9 days. A worked alone for 8 days then B worked alone for 10 days 
and finished the project. In how many days can A complete the project alone? B?
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<pre>
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{{{1/9}}} = {{{73/9}}}

    a + b = {{{9/73}}}.            (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 = {{{72/73}}}          (3)

    8a + 10b = 1             (4)


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

          10b - 2b = 1 - {{{72/73}}},   or

           2b      = {{{73/73 - 72/73}}} = {{{1/73}}}.


Hence,  b = {{{1/(2*73)}}} = {{{1/146}}}.

It means that B makes  {{{1/146}}}  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 - {{{10/146}}} = {{{136/146}}};

hence,  a = {{{136/146}}} : 8 = {{{17/146}}}.


It means that A needs  {{{146/17}}} = 8 {{{10/17}}}  days to complete the job alone.
</pre>

Solved.


The numbers in the answer are ugly &nbsp;(at least one is ugly), &nbsp;but I checked them against the equations &nbsp;(1) &nbsp;and &nbsp;(2), 
and the answer is &nbsp;CORRECT.