Question 1168985
.
A t-shirt design company is dying purple shirts for a school club. The shirts came out too light so the company needs to add 
more pure blue dye to the mixture to get a darker purple color for the shirts. The full vat of color holds 54 gallons 
of the original dye that is 27% blue. How many gallons should be drained and replaced with pure blue dye so that the mixture will be 40% blue?
~~~~~~~~~~~~~



<pre>
Let V be the volume to drain off from 54 gallons of the 27% solution.


<U>Step 1:  Draining</U>.  After draining,  you have (54-V) gallons of the 27% acid solution.



<U>Step 2:  Replacing</U>.  Then you add V gallons of the pure blue dye (the replacing step).

                     After the replacing,  you have the same total liquid volume of 54 gallons.


It contains  0.27*(54-V) + V   of pure blue dye.



So, your "concentration equation" is


    {{{(0.27*(54-V) + V)/54}}} = 0.4.    (1)



At this point, the setup is done and completed.



Now you need to solve your basic equation  (1).

As the first step, multiply both sides by 80, and then simplify


    0.27*(54-V) + V = 0.4*54

    14.58 - 0.27V + V = 21.6

    0.73V = 21.6 - 14.58 = 7.02

    V = {{{7.02/0.73}}} = 9.616.


<U>ANSWER</U>.  9.616 gallons of the 27% solution should be drained and replaced by the pure blue duy.


<U>CHECK</U>.   {{{(0.27*(54-9.616) + 9.616)/54}}} = 0.4.    ! Precisely correct !
</pre>

Solved.


---------------------


There is entire bunch of lessons covering various types of mixture problems

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/mixtures/Mixture-problems.lesson>Mixture problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/mixtures/More-Mixture-problems.lesson>More Mixture problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/mixtures/Solving-typical-mixture-problems.lesson>Solving typical word problems on mixtures for solutions</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/Word-problems-on-mixtures-for-antifreeze-solutions.lesson>Word problems on mixtures for antifreeze solutions</A> (*)

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/mixtures/Solving-typical-word-problems-on-mixtures-for-dry-substances-coffee-nuts.lesson>Word problems on mixtures for dry substances like coffee beans, nuts, cashew and peanuts</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/mixtures/Word-problems-on-mixtures-for-dry-substances-like-cookies-candies.lesson>Word problems on mixtures for dry substances like candies, dried fruits</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/mixtures/Word-problems-on-mixtures-for-dry-substances-like-soil-and-sand.lesson>Word problems on mixtures for dry substances like soil and sand</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/Word-problems-on-mixtures-for-alloys.lesson>Word problems on mixtures for alloys</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/Typical-word-problems-on-mixtures-from-the-archive.lesson>Typical word problems on mixtures from the archive</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/Advanced-mixture-problems.lesson>Advanced mixture problems</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/mixtures/Advanced-mixture-problem-for-three-alloys.lesson>Advanced mixture problem for three alloys</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/mixtures/OVERVIEW-of-lesson-on-word-problems-for-mixtures.lesson>OVERVIEW of lessons on word problems for mixtures</A>

in this site.


A convenient place to quickly observe these lessons from a &nbsp;"bird flight height" &nbsp;(a top view) &nbsp;is the last lesson in the list.


Read them and become an expert in solution the mixture word problems.


Specially for "draining - replacing" mixture problems see the lesson marked (*) in the list.



Also, you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic "<U>Mixture problems</U>".



Save the link to this online textbook together with its description


Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson


to your archive and use it when it is needed.