Question 1157110
You are given their rates of working:
Anna: [ 1 job ] / [ 12 days ]
Beth: [ 1 job ] / [ 15 days ]
Charles [ 1 job ] / [ 30 days ]
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Anna starts working alone and
works for 3 days. In 3 days she
does this fraction of the job:
{{{ ( 1/12 )*3 = 1/4 }}} 
There is {{{ 3/4 }}} of the job left
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Anna and Beth then work together for
an unknown number of days. 
Let {{{ d }}} = the number of days they work together
Their rate of working together is:
{{{ 1/12 + 1/15 = 5/60 + 4/60 }}}
{{{ 1/12 + 1/15 = 9/60 }}}
{{{ 9/60 = 3/20 }}}
The fraction of the remaining work they get done is:
{{{ ( 3/20 )*d*(3/4) = ( 9/80 )*d }}}
So now {{{ 1/4 + ( 9/80 )*d  }}} is done
{{{ 1 - 1/4 - ( 9/80 )*d = 3/4 - ( 9/80 )*d }}} is
the fraction left to do
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Now for 3 days, Charles and Beth work together
and they finish the job
Add their rates of working
{{{ 1/15 + 1/30 = ( 3/4 - ( 9/80 )*d ) / 3 }}}
{{{ 1/15 + 1/30 = 1/4 - ( 3d ) / 80 }}}
{{{ 2/30 + 1/30 = 20/80 - (3d)/80 }}}
{{{ 3/30 = ( 20 - 3d ) / 80 }}}
{{{ 1/10 = ( 20 - 3d )/ 80 }}}
Multiply both sides by {{{ 80 }}}
{{{ 8 = 20 - 3d }}}
{{{ 3d = 12 }}}
{{{ d = 4 }}}
So Anna and Beth worked together for {{{ 4 }}} days
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Add up the days they all worked:
Anna alone: 3 days
Anna & Beth: 4 days
Charles and Beth: 3 days
{{{ 3 + 4 + 3 = 10 }}}
They finish the work in 10 days
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Check my math and get a 2nd opinion if needed