Question 206874
Let 

x = volume of the first solution (in cubic centimeters)
y = volume of the mixed solution. In other words, the volume of the combination of the two volumes of the 'x' and 10 cubic centimeter solutions. (note: volume is again in cubic centimeters)




Since 'y' is the combination of the first two solutions, this means that {{{y=x+10}}}



To find the second equation, this is where things get a bit tricky. There are 'x' cubic centimeters of the 80% solution (of HCI). So this means that there are 0.8x cubic centimeters of pure HCI in the first solution. Likewise, in the second solution, there are 10 cubic centimeters. Since the second solution is 20% HCI, this tells us that there is 0.2(10) cubic centimeters of pure HCI in the second solution. Now since we want the final solution to be 60% HCI, this means we're going to set the sum of 0.8x and 0.2(10) equal to 0.6y (since 0.6y is also pure HCI). 



In other words, we're going to have this second equation: {{{0.8x+0.2(10)=0.6y}}}



{{{0.8x+0.2(10)=0.6y}}} Start with the given equation.



{{{0.8x+0.2(10)=0.6(x+10)}}} Plug in {{{y=x+10}}} (the first equation)



{{{0.8x+0.2(10)=0.6x+0.6(10)}}} Distribute



{{{0.8x+2=0.6x+6}}} Multiply



{{{8x+20=6x+60}}} Multiply EVERY term by 10 to move the decimal points one spot to the right. This will make every value a whole number.



{{{8x=6x+60-20}}} Subtract {{{20}}} from both sides.



{{{8x-6x=60-20}}} Subtract {{{6x}}} from both sides.



{{{2x=60-20}}} Combine like terms on the left side.



{{{2x=40}}} Combine like terms on the right side.



{{{x=(40)/(2)}}} Divide both sides by {{{2}}} to isolate {{{x}}}.



{{{x=20}}} Reduce.



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Answer:


So the solution is {{{x=20}}}



This means that 20 cubic centimeters of the first solution (that is 80% HCI) is needed.