Question 1208723
.
Ruth has a beaker containing a solution of 800 mL of acid and 200 mL of water.  
She thinks the solution is a little strong, so she drains 300 mL from the beaker, adds 300 mL of water, and stirs the solution.  
Ruth thinks the solution is still too strong, so again she drains 400 mL from the beaker, and adds 400 mL of water.  
Ruth again thinks the solution is still too strong, so again she drains 100 mL from the beaker, and adds 100 mL of water.  
How many mL of water are now in the beaker?
~~~~~~~~~~~~~~~~~~~~~~



        The answers in the posts by  Edwin and by  @MathThearapy are different - so,  there is the need to check their solutions.


        Here I make this check, but I use slightly different methodology, which better suits to the problem.



        In this problem, after each step, Ruth keeps the total volume of the mixture in the beaker unchangeable, 1000 mL.


        Therefore, in my solution, I calculate and watch/track for the concentration/amount of the acid, only, in the beaker.


        I will calculate the amount of water in the beaker after the last step, but will not calculate/watch/track it at the intermediate steps.


        It will allow me to reduce the volume of calculations and will allow me to concentrate

        my attention on one component, only (which is acid). It will diminish the volume 

        of calculations and will diminish possible errors.



<pre>
(0)  Starting amount of the acid is 800 mL;  the starting concentration is  {{{800/1000}}} = 0.8.



(1)  First step is to drain 300 mL of the mixture from the beaker.

     With it,  0.8*300 = 240 mL of the acid goes out.

     The amount of the acid remained in the beaker is 800-240 = 560 mL.

     The amount of water is added to keep the total volume of the mixture in the beaker 1000 mL.

     The concentration after step 1  is  {{{560/1000}}} = 0.56.



(2)  Second step is to drain 400 mL of the mixture from the beaker.

     With it,  0.56*400 = 224 mL of the acid goes out.

     The amount of the acid remained in the beaker is 560-224 = 336 mL.

     The amount of water is added to keep the total volume of the mixture in the beaker 1000 mL.

     The concentration after step 2  is  {{{336/1000}}} = 0.336.



(3)  Third step is to drain 100 mL of the mixture from the beaker.

     With it,  0.336*100 = 33.6 mL of the acid goes out.

     The amount of the acid remained in the beaker is 336-33.6 = 302.4 mL.

     The amount of water is added to keep the total volume of the mixture in the beaker 1000 mL.

     
     Hence, the amount of water in the beaker is  1000 - 302.4 = 697.6 mL.


<U>ANSWER</U>.  The amount of water in the beaker after step 3 is  697.6 mL.
</pre>

Solved.


The answer by @MathTherapy is confirmed. 


Calculations are made by the most economic/effective way.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Of course, &nbsp;the most important element of my post 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;is the &nbsp;METHODOLOGY, &nbsp;which &nbsp;EXCLUDES &nbsp;unnecessary calculations.



The answer by Edwin is incorrect.



//////////////////////////////



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;To help &nbsp;Edwin to identify his error, &nbsp;I copied and pasted here his solution

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; from his post, &nbsp;and pointed the precise place/line with the error.



<pre>
Ruth has a beaker containing a solution of 800 mL of acid and 200 mL of water.
She thinks the solution is a little strong, so she drains 300 mL from the
beaker, adds 300 mL of water, and stirs the solution. Ruth thinks the solution
is still too strong, so again she drains 400 mL from the beaker, and adds 400 mL
of water. Ruth again thinks the solution is still too strong, so again she
drains 100 mL from the beaker, and adds 100 mL of water. How many mL of water
are now in the beaker?

I'll keep track of the liquid, acid, water, % acid, and % water, because in
other problems like this you might be asked different amounts and percentages.
 
<font color="red"><b>Ruth has a beaker containing a solution of 800 mL of acid and 200 mL of water.</b></font> 

 liquid = 800+200 = 1000 mL
   acid = 800 mL
  water = 200 mL
 % acid = 800/1000 = 0.8 = 80%
% water = 200/1000 = 0.2 = 20% 

<font color="red"><b>She thinks the solution is a little strong, so she drains 300 mL from the beaker,</b></font>  

 liquid =  1000-300 = 700 mL
   acid =  (0.8)(700) = 560 mL
  water =  (0.2)(700) = 140 mL
 % acid =   80%
% water =   20% 

<font color="red"><b>adds 300 mL of water, and stirs the solution.</b></font>  

 liquid =  700+300 = 1000 mL
   acid =  560 mL
  water =  140+300 = 440 mL
 % acid =  560/1000 = 0.56 = 56%
% water =  440/1000 = 0.44 = 44% 

<font color="red"><b>Ruth thinks the solution is still too strong, so again she drains 400 mL from the beaker,</b></font>  

 liquid =  1000-400 = 600 mL
   acid =  (600)(0.56) = 336 mL
  water =  (600)(0.44) = 264 mL
 % acid =  56%
% water =  44% 

<font color="red"><b>and adds 400 mL of water.</b></font>  

 liquid =  600+400=1000 mL
   acid =  336 mL
  water =  264+400=664 mL
 % acid =  336/1000=33.6% 
% water =  664/1000=66.4% 


<font color="red"><b>Ruth again thinks the solution is still too strong, so again she drains 100 mL from the beaker,</b></font>  

 liquid =  1000-100=900 mL
   acid =  (900)(0.336) = 302.4 mL   <<<---===  The error is HERE.  The correct calculation should be acid = 1000*0.336 = 336 mL, 
                                                                    with relevant correction in  all consequential calcs. 
  water =  664-100=564 mL
 % acid =  33.6% 
% water =  66.4% 

<font color="red"><b>and adds 100 mL of water.</b></font>  

 liquid =  900+100=1000 mL 
   acid =  302.4 mL
  water =  564+100=664 mL
 % acid =  302.4/1000 0.3024=30.24% 
% water =  664/1000 = 0.664 = 66.4% 

<font color="red"><b>How many mL of water are now in the beaker?</b></font> 

664 mL   <-- ANSWER

Edwin</pre>

...................................



This shows once again how dangerous is to split your attention 
by making unnecessary calculations.   


Professional writers in such cases say: how good that this mistake was made naturally; 
otherwise, it would have been worth to make it intentionally, and then talking, discussing  
and teaching on how to organize calculations in a secure way.



&nbsp;&nbsp;&nbsp;&nbsp;@ikleyn