Question 1150928
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The first and the major step in solving such problems is to understand what is a 10% salt solution  and what is a 3% salt solution.


The concentration in such problems is measured in kilograms of salt per liter of the mixture.


A 10% salt solution is 1 liter of water with dissolved 100 grams of salt in it - or any proportional mixture.    (1)


A 3% salt solution is 1 liter of water with dissolved 30 grams of salt in it - or any proportional mixture.      (2)



    These are not unique possible definitions.

    One can define a 10% salt solution as a mixture that contains 10% of salt by weight (by total mass of the solution).

    It is not exactly the same definition as (1) - these two definitions are <U>slightly different</U>.

    But in school Math, students and teachers usually NEGLECT this difference,
    and consider and accept (1) and (2) as the basic definitions.


So, taking into account that 1 gallon = 3.785 liters, we can say that 45 gallons of 3% salt solution contain

      45*3.785*0.03 kilograms of salt.    (3)


Since we obtained these 45 gallons of the 3% salt solution by adding water,
our original 10% salt solution contained exactly this (= the same) amount (3) of salt.


Hence, the volume of the original 10% salt solution was 

     volume = {{{salt_amount/salt_concentration}}} = {{{(45*3.785*0.03)/0.1}}} = 45*3.785*0.3 liters.


It means that the volume of water added is the difference

    volume of water added = 45*3.785 - 45*3.785*0.3 liters = 45*3.785*(1-0.3) = 45*3.785*0.7 liters.    (4)



If you want to express this volume in gallons, you must divide (4) by the conversion factor of 3.785.


Thus the volume of water to add in gallons is  45*0.7 = 31.5 gallons.    <U>ANSWER</U>
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Solved.


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For introductory lessons covering various types of mixture word problems see

&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/Typical-word-problems-on-mixtures-from-the-archive.lesson>Typical word problems on mixtures from the archive</A>

in this site.


You will find there ALL TYPICAL mixture problems with different methods of solutions,
explained at different levels of detalization, &nbsp;from very detailed to very short.


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


Also, &nbsp;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 textbook in the section "<U>Word problems</U>" 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.