A chemist needs to mix a solution containing 30% insecticide with a solution containg 50% insecticide to make 200L of a solution that 42% insecticide. How much of each solution should she use?
dealing with- graphing, subst., and elimin.
let x L represent the volume of the 30% solution, then 200 - x L would be the 50% solution
then...
.3x + .5(200-x) = .42(200)
solve....
To solve this problem, we can use a method called the method of substitution or elimination. However, graphing is not necessary in this case.
Let's assume the chemist uses x liters of the 30% insecticide solution and y liters of the 50% insecticide solution to make a total of 200L of 42% insecticide solution.
We can set up two equations based on the given information:
1) The total volume equation:
x + y = 200
2) The concentration equation:
0.30x + 0.50y = 0.42(200)
Now, we can proceed using either substitution or elimination method.
Substitution method:
1) Solve the first equation for x:
x = 200 - y
2) Substitute x in the second equation:
0.30(200 - y) + 0.50y = 0.42(200)
60 - 0.30y + 0.50y = 84
0.20y = 24
y = 120
3) Substitute y back into the first equation to find x:
x + 120 = 200
x = 80
Therefore, the chemist should use 80 liters of the 30% insecticide solution and 120 liters of the 50% insecticide solution to make 200 liters of a 42% insecticide solution.
Elimination method:
1) Multiply the first equation by -0.30:
-0.30x - 0.30y = -60
2) Multiply the second equation by 100 to get rid of decimals:
30x + 50y = 84(200)
30x + 50y = 16800
3) Add the two equations together:
-0.30x - 0.30y + 30x + 50y = -60 + 16800
29.70x + 49.70y = 16740
4) Divide the equation by 0.10 to eliminate decimals:
297x + 497y = 167400
5) Multiply the first equation by -497:
-497x - 497y = -99400
6) Add the equations together:
297x + 497y - 497x - 497y = 167400 - 99400
-200x = 68000
x = -340
Since we cannot have a negative volume, we discard this solution. Hence, the elimination method does not yield a valid answer in this case.
Therefore, the chemist should use 80 liters of the 30% insecticide solution and 120 liters of the 50% insecticide solution to make 200 liters of a 42% insecticide solution.