Question 767278
Look at the cold & hot combined rate.
In units of tub per minute, you have {{{1/10+1/15=1/6}}} tub per minute, not here showing that arithmetic process.  The EXPECTED length of time to fill the tub was 6 minutes, so this person let the faucets fill for 6 minutes.


BUT the drain was also open, its rate an unknown, r.


Let us use {{{R*m=v}}} where R is rate in tubs or vessel-fulls per minute, m is minutes, and v is vessel-fulls, in this case, meaning tubs.  Also think clearly about the variables.  R is for "in general" and r is for our specific tub's drain pipe.  Also be aware, {{{r<0}}}.   r is negative.


So we have two parts of the job.  One part when all three drains were open for 6 minutes:  {{{(1/6+r)*6}}}
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and the part when our person closed the drain pipe and the tub continued to fill for 4 minutes:  {{{(1/6)*4}}}


The sum of those two expressions is the quantity of 1 tub.
{{{highlight((1/6+r)*6+(1/6)*4=1)}}}
SOLVE FOR r.  That is the rate of draining the tub for the drain pipe.