document.write( "Question 24136: How many gallons of a 70% salt solution must be mixed with 60 gallons of a 18% solution to obtain a solution that is 60% salt? \n" ); document.write( "

Algebra.Com's Answer #12823 by kev82(151) $\"\"$ $\"About$

You can put this solution on YOUR website!

Call the 70% solution A, and the 18% solution B

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Solution B provides you with $\"0.18%2A60=10.8\"$ units of salt in 60 units. For every unit of solution A you add, you get 0.7 units of salt. This is just the definition of percentages.

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So, when adding x units of solution A we get $\"10.8+%2B+0.7x\"$ units of salt, and we have $\"60%2Bx\"$ gallons, so the percentage salt in the finished solution is:

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\n" ); document.write( "\n" ); document.write( " $\"%2810.8%2B0.7x%29%2F%2860%2Bx%29\"$ \r
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We want this to equal 60%, or 0.6, so we have to solve the equation:

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\n" ); document.write( "\n" ); document.write( " $\"%2810.8%2B0.7x%29%2F%2860%2Bx%29=0.6\"$ \r
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Unfortunately, I'm not clever enough to make the automatic solver, solve that. But if you multiply both sides by $\"60%2Bx\"$ and then put it through the automatic solver it works. See below:

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Solved by pluggable solver: EXPLAIN simplification of an expression

Your Result:

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This is an equation! Solutions: x=252.
Graphical form: Equation $\"10.8%2B0.7%2Ax=36%2B0.6%2Ax\"$ was fully solved.
Text form: 10.8+0.7*x=36+0.6*x simplifies to 0=0
Cartoon (animation) form: $\"simplify_cartoon%28+10.8%2B0.7%2Ax=36%2B0.6%2Ax+%29\"$
For tutors: simplify_cartoon( 10.8+0.7*x=36+0.6*x )
If you have a website, here's a link to this solution.

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DETAILED EXPLANATION

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Look at $\"highlight_red%28+10.8+%29%2B0.7%2Ax=36%2B0.6%2Ax\"$ .
Moved $\"10.8\"$ to the right of expression
It becomes $\"0.7%2Ax%2Bhighlight_green%28+10.8+%29=36%2B0.6%2Ax\"$ .
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Look at $\"0.7%2Ax%2B10.8=highlight_red%28+36+%29%2B0.6%2Ax\"$ .
Moved $\"36\"$ to the right of expression
It becomes $\"0.7%2Ax%2B10.8=0.6%2Ax%2Bhighlight_green%28+36+%29\"$ .
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Look at $\"0.7%2Ax%2B10.8=highlight_red%28+0.6%2Ax%2B36+%29\"$ .
Moved these terms to the left $\"highlight_green%28+-0.6%2Ax+%29\"$ , $\"highlight_green%28+-36+%29\"$
It becomes $\"0.7%2Ax%2B10.8-highlight_green%28+0.6%2Ax+%29-highlight_green%28+36+%29=0\"$ .
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Look at $\"0.7%2Ax%2Bhighlight_red%28+10.8+%29-0.6%2Ax-highlight_red%28+36+%29=0\"$ .
Added fractions or integers together
It becomes $\"0.7%2Ax-highlight_green%28+252%2F10+%29-0.6%2Ax=0\"$ .
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Look at $\"0.7%2Ax-highlight_red%28+252%2F10+%29-0.6%2Ax=0\"$ .
Moved $\"-252%2F10\"$ to the right of expression
It becomes $\"0.7%2Ax-0.6%2Ax-highlight_green%28+252%2F10+%29=0\"$ .
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Look at $\"0.7%2Ax-0.6%2Ax-highlight_red%28+252+%29%2Fhighlight_red%28+10+%29=0\"$ .
Factors 252 and 10 have greatest common factor (GCF) of 2. Reducing fraction.
It becomes $\"0.7%2Ax-0.6%2Ax-highlight_green%28+126+%29%2Fhighlight_green%28+5+%29=0\"$ .
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Look at $\"highlight_red%28+0.7%2Ax+%29-highlight_red%28+0.6%2Ax+%29-126%2F5=0\"$ .
Eliminated similar terms $\"highlight_red%28+0.7%2Ax+%29\"$ , $\"highlight_red%28+-0.6%2Ax+%29\"$ replacing them with $\"highlight_green%28+%280.7-0.6%29%2Ax+%29\"$
It becomes $\"highlight_green%28+%280.7-0.6%29%2Ax+%29-126%2F5=0\"$ .
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Look at $\"%28highlight_red%28+0.7+%29-highlight_red%28+0.6+%29%29%2Ax-126%2F5=0\"$ .
Added fractions or integers together
It becomes $\"%28highlight_green%28+1%2F10+%29%29%2Ax-126%2F5=0\"$ .
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Look at $\"highlight_red%28+%28highlight_red%28+1%2F10+%29%29%2Ax+%29-126%2F5=0\"$ .
Remove unneeded parentheses around factor $\"highlight_red%28+1+%29\"$ , $\"highlight_red%28+1%2F10+%29\"$
It becomes $\"highlight_green%28+1+%29%2Fhighlight_green%28+10+%29%2Ax-126%2F5=0\"$ .
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Look at $\"highlight_red%28+1+%29%2F10%2Ax-126%2F5=0\"$ .
Remove extraneous '1' from product $\"highlight_red%28+1+%29\"$
It becomes $\"x%2F10-126%2F5=0\"$ .
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Look at $\"highlight_red%28+x%2F10-126%2F5+%29=0\"$ .
Solved linear equation $\"highlight_red%28+x%2F10-126%2F5=0+%29\"$ equivalent to 0.1*x-25.2 =0
It becomes $\"highlight_green%28+0+%29=0\"$ .
Result: $\"0=0\"$
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Universal Simplifier and Solver

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Done!

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And you get the answer 252. Don't forget to check it by putting your answer back into the equation!

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Hope this helps

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Kev

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