Question 200265
Let 


x = amount of brine solution with 1% salt (in liters)
y = amount of brine solution with 4% salt (in liters)



Since he's mixing the two solutions to get one liter, this means that {{{x+y=1}}}. In other words, the two solutions add up to one liter.



So the first equation is {{{x+y=1}}}



Since "x" is the amount of the 1% solution, this means that {{{0.01x}}} is the amount of pure salt (since 1% of the given solution is given to be salt). Also, {{{0.04y}}} is the amount of pure salt (from the other solution). These figures add up to the total {{{0.028(x+y)}}} (note: 0.028 is the percentage while x+y is the total amount). Now add up the two parts and set them equal to the last portion to get {{{0.01x+0.04y=0.028(x+y)}}}



{{{0.01x+0.04y=0.028(x+y)}}} Start with the given equation.



{{{0.01x+0.04y=0.028x+0.028y}}} Distribute



{{{10x+40y=28x+28y}}} Multiply EVERY number by 1000 (to move the decimal 3 spots to the right)



{{{10x+40y-28x-28y=0}}} Get everything to one side



{{{-18x+12y=0}}} Combine like terms.



So the second equation is {{{-18x+12y=0}}}





So we have the system of equations:


{{{system(x+y=1,-18x+12y=0)}}}



{{{18(x+y)=18(1)}}} Multiply the both sides of the first equation by 18.



{{{18x+18y=18}}} Distribute and multiply.



So we have the new system of equations:


{{{system(18x+18y=18,-18x+12y=0)}}}



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:



{{{(18x+18y)+(-18x+12y)=(18)+(0)}}}



{{{(18x+-18x)+(18y+12y)=18+0}}} Group like terms.



{{{0x+30y=18}}} Combine like terms.



{{{30y=18}}} Simplify.



{{{y=(18)/(30)}}} Divide both sides by {{{30}}} to isolate {{{y}}}.



{{{y=3/5}}} Reduce.



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{{{18x+18y=18}}} Now go back to the first equation.



{{{18x+18(3/5)=18}}} Plug in {{{y=3/5}}}.



{{{18x+54/5=18}}} Multiply.



{{{5(18x+54/cross(5))=5(18)}}} Multiply both sides by the LCD {{{5}}} to clear any fractions.



{{{90x+54=90}}} Distribute and multiply.



{{{90x=90-54}}} Subtract 54 from both sides.



{{{90x=36}}} Combine like terms.



{{{x=36/90}}} Divide both sides by {{{54}}} to isolate {{{x}}}.



{{{x=2/5}}} Reduce.



So the solutions are {{{x=2/5}}} and {{{y=3/5}}}.



Which form the ordered pair *[Tex \LARGE \left(\frac{2}{5},\frac{3}{5}\right)].



These solutions in decimal form are {{{x=0.4}}} and {{{y=0.6}}}



Now recall that we stated that the values of "x" and "y" are in units of liters. So this means that x=0.4 liters and y=0.6 liters



Multiply both values by 1000 to convert to milliliters: 1000*0.4=400, 1000*0.6=600



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Answer:



So this means that 400 milliliters of the 1% salt solution and 600 milliliters of the 4% salt solution are needed to make a 1 L solution that is 2.8% salt.