Question 830688
Man's rate, 1/y
Woman's rate , 1/x


{{{4(1/y)+6(1/x)=1/8 }}}
and
{{{3(1/y)+7(1/x)=1/10}}}


Simplify the first equation:
4/y+6/x=1/8, LCD 8xy
32x+48y=xy
Simplify the second equation:
3/y+7/x=1/10, LCD 10xy
30x+70y=xy


Find the system:
32x+48y=xy and 30x+70y=xy.
Subtracting the first one from the second one eliminates the xy term, member.
-2x+22y=0
-x+11y=0
{{{highlight_green(11y=x)}}}


Try substituting this in one of the equations of the system and use this to solve for y:
30x+70y=xy
30(11y)+70y=(11y)y
330y+70y=11y^2
400y=11y^2
11y^2-400y=0
{{{highlight_green(y(11y-400)=0)}}}
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We can get a value for y from that.  The y=0 is useless, but the 11y-400=0 gives {{{highlight(y=400/11)}}};
Use this to find x.
-
30x+70y=xy
30x+79(400/11)=x(400/11)
(400/11)x-30x=79(400/11)
Multiply left and right by 11
400x-330x=79*400
70x=79*400
7x=79*40
x=79*40/7
{{{highlight(x=451)}}}, and y=36


How many days if just 10 women work?
{{{10(1/451)*d=1}}}   for the one job, d for how many days.
{{{d=451/10}}}
{{{highlight(d=45.1)}}} days