document.write( "Question 1154197: A chemical company makes two brands of antifreeze. The first brand is 45% pure antifreeze, and the second brand is 85% pure antifreeze. In order to obtain 40 gallons of a mixture that contains 75% pure antifreeze, how many gallons of each brand of antifreeze must be used? \n" ); document.write( "
Algebra.Com's Answer #776605 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "            Antifreeze is a mixture of water with some liquid chemical - ethylene glycole or propylene glycole.\r
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\n" ); document.write( "\n" ); document.write( "            For purposes of this problem,  you don't need know exactly,  what these chemicals are.\r
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\n" ); document.write( "\n" ); document.write( "            It is quite enough to know that antifreeze is a mixture of water with some liquid chemical.\r
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\n" ); document.write( "\n" ); document.write( "            In school math,  there are two major approaches/ways to solve such problems.\r
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\n" ); document.write( "\n" ); document.write( "            In my post,  I will show you one of them --- using two-equations setup.\r
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document.write( "Let x be the volume of the first brand (in gallons), and y be the volume of the second brand.\r\n" );
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document.write( "Then the total volume of the mixture is (x+y) gallons.\r\n" );
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document.write( "First  brand contains 45% of the antifreeze, which is 0.45x gallons of pure antifreeze.\r\n" );
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document.write( "Second brand contains 85% of the antifreeze, which is 0.85x gallons of pure antifreeze.\r\n" );
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document.write( "Therefore, from the condition, you have these two equations\r\n" );
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document.write( "    x + y = 40   gallons               (1)    \r\n" );
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document.write( "    \"%280.45x+%2B+0.85y%29%2F40\" = 0.75  gallons       (2)   \r\n" );
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document.write( "First equation is for the total volume of the mixtures.\r\n" );
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document.write( "In second equation, the numerator is the volume of the pure antifreeze.\r\n" );
document.write( "Divided by the denominator (40 gallons), it is the concentration of the resulting mixture, which is given as 75%, \r\n" );
document.write( "by the condition.\r\n" );
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document.write( "Thus (1) and (2) are two basic equations to solve the problem.\r\n" );
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document.write( "Taken together, they form the setup for the given problem. They are the base to solve the given problem.\r\n" );
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document.write( "This setup is called \"2-equations setup\".\r\n" );
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document.write( "If you understand, how these equations are constructed, why they have their form, then you just understand the problem.\r\n" );
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document.write( "Now our task is to solve the system of equations.\r\n" );
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document.write( "To do it, first multiply both sides of the equation (2) by 40. You will get then this system\r\n" );
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document.write( "    x + y = 40                   (3)    \r\n" );
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document.write( "    0.45x + 0.85y = 0.75*40      (4)   \r\n" );
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document.write( "Next, from equation (3), express x = 40-y  and substitute it into equation (4). You will get\r\n" );
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document.write( "    0.45*(40-y) + 0.85*y = 0.75*40\r\n" );
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document.write( "    -0.45y      + 0.85y  = 0.75*40 - 0.45*40\r\n" );
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document.write( "      y                  = \"%280.75%2A40-+0.45%2A40%29%2F%280.85-0.45%29\" = 30.\r\n" );
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document.write( "So, 30 gallons of the second brand should be used.\r\n" );
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document.write( "Then it is clear (or derive it from equation (1)) that the volume of the first brand should be 40-30 = 10 gallons.\r\n" );
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document.write( "ANSWER.  10 gallons of the first brand and 30 gallons of the second brand.\r\n" );
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document.write( "CHECK.    There is no need to check equation (1), since right now we derived the first brand volume from it.\r\n" );
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document.write( "          In reality, we should check equation (2) only :  \"%280.45%2A10+%2B+0.85%2A30%29%2F40\" = 0.75 = 75%.   ! Precisely correct !\r\n" );
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\n" ); document.write( "\n" ); document.write( "As an after-solution conclusion, I want to make a notice.\r
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document.write( "    In the solution process, I did not make each single calculation separately.\r\n" );
document.write( "    I created the final formula, and then used Excel in my computer.\r\n" );
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document.write( "    I don't know, which calculation tool do you have: pocket calculator or some software on your comp.\r\n" );
document.write( "    I don't know which calculation tool you are assigned to use.\r\n" );
document.write( "    But I used to deduce all my calcs to one formula (when possible); then I copy the formula into a cell in Excel \r\n" );
document.write( "    spreadsheet and get the answer in one click (and with no errors).   Again: in one click with no errors (!)\r\n" );
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document.write( "    You may perform differently - I prefer do not submerge in these details.\r\n" );
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document.write( "    Same way I make my check: I write formula, then copy and paste it into Excel cell and then get an answer in one click.\r\n" );
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document.write( "    Therefore, all talks if it is economic solution path from the calculation point of view or not - are out me: \r\n" );
document.write( "    it is not my goal to discuss it with the student at this level / (at this stage).\r\n" );
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document.write( "    At this stage, my goal is to teach on how to solve the problem by Algebra methods.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Now,  what I wrote above,  is appropriate to teach the student,  when he or she solves the problem first time in his or her life.\r
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\n" ); document.write( "\n" ); document.write( "When the student will solve such problems 2-3 times,  he  (or she)  will be inclined to write the solution in much shorter way.\r
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\n" ); document.write( "\n" ); document.write( "Now I want to present you this short way.\r
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document.write( "Let x be the volume of the first brand (in gallons), and y be the volume of the second brand.\r\n" );
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document.write( "Then from the condition, you have these two equations\r\n" );
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document.write( "    x + y = 40   gallons           (1)   (total volume equation)\r\n" );
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document.write( "    \"%280.45x+%2B+0.85y%29%2F40\" = 0.75            (2)   (the concentration equation)\r\n" );
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document.write( "You can write equation (2) in the form\r\n" );
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document.write( "     0.45x + 0.85y = 0.75*40\r\n" );
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document.write( "Next, from equation (3), express x = 40-y  and substitute it into equation (4). You will get\r\n" );
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document.write( "    0.45*(40-y) + 0.85*y = 0.75*40\r\n" );
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document.write( "From this equation, express y and calculate\r\n" );
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document.write( "      y                  = \"%280.75%2A40-+0.45%2A40%29%2F%280.85-0.45%29\" = 30.\r\n" );
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document.write( "So, 30 gallons of the second brand should be mixed with 40-30 = 10 gallons of the first brand.    ANSWER\r\n" );
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document.write( "CHECK.   We should check equation (2) only :  \"%280.45%2A10+%2B+0.85%2A30%29%2F40\" = 0.75 = 75%.   ! Precisely correct !\r\n" );
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\n" ); document.write( "\n" ); document.write( "It is the short form solution.\r
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\n" ); document.write( "\n" ); document.write( "There is the shortest form,  too,  when all explanations are omitted,  as follows:\r
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document.write( "    x + y = 40   gallons       (total volume equation)\r\n" );
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document.write( "    \"%280.45x+%2B+0.85y%29%2F40\" = 0.75         (the concentration equation)\r\n" );
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document.write( "     0.45x + 0.85y = 0.75*40\r\n" );
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document.write( "    0.45*(40-y) + 0.85*y = 0.75*40\r\n" );
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document.write( "      y                  = \"%280.75%2A40-+0.45%2A40%29%2F%280.85-0.45%29\" = 30.\r\n" );
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document.write( "So, 30 gallons of the second brand should be mixed with 40-30 = 10 gallons of the first brand.    ANSWER\r\n" );
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document.write( "CHECK.   \"%280.45%2A10+%2B+0.85%2A30%29%2F40\" = 0.75 = 75%.   ! Precisely correct !\r\n" );
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\n" ); document.write( "\n" ); document.write( "So,  only  7  lines are required for the solution,  including one line for the answer and  1  line for the check -- when you know the method.\r
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document.write( "    Interesting, that there is even shorter way, when you use ONE-EQUATION setup.\r\n" );
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document.write( "    Then two lines are totally enough to place your solution.\r\n" );
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document.write( "    But since my post is already too long, I can tell you about it next time, if you ask me about it.\r\n" );
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document.write( "    Or, alternatively, you may learn it from the links that follow.\r\n" );
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\n" ); document.write( "\n" ); document.write( "It is a standard and typical mixture problem.\r
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\n" ); document.write( "\n" ); document.write( "There is entire bunch of lessons covering various types of mixture problems\r
\n" ); document.write( "\n" ); document.write( "    - Mixture problems\r
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\n" ); document.write( "\n" ); document.write( "    - Solving typical word problems on mixtures for solutions \r
\n" ); document.write( "\n" ); document.write( "    - Word problems on mixtures for antifreeze solutions (*)\r
\n" ); document.write( "\n" ); document.write( "    - Word problems on mixtures for dry substances like coffee beans, nuts, cashew and peanuts\r
\n" ); document.write( "\n" ); document.write( "    - Word problems on mixtures for dry substances like candies, dried fruits\r
\n" ); document.write( "\n" ); document.write( "    - Word problems on mixtures for dry substances like soil and sand\r
\n" ); document.write( "\n" ); document.write( "    - Word problems on mixtures for alloys \r
\n" ); document.write( "\n" ); document.write( "    - Typical word problems on mixtures from the archive\r
\n" ); document.write( "\n" ); document.write( "    - Advanced mixture problems \r
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\n" ); document.write( "\n" ); document.write( "    - Check if you know the basics of mixtures from Science \r
\n" ); document.write( "\n" ); document.write( "    - OVERVIEW of lessons on word problems for mixtures\r
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\n" ); document.write( "\n" ); document.write( "Among them, the lesson marked (*) in the list is specially devoted to antifreeze mixtures.\r
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\n" ); document.write( "\n" ); document.write( "A convenient place to quickly observe these lessons from a  \"bird flight height\"  (a top view)  is the last lesson in the list.\r
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\n" ); document.write( "\n" ); document.write( "Read them and become an expert in solution the mixture word problems.\r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic \"Mixture problems\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I
\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "to your archive and use it when it is needed.\r
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