A mixture of 12 gallons of chemical A, 16 gallons of chemical B, and 26 gallons of chemical C is required.....

Question

A mixture of 12 gallons of chemical A, 16 gallons of chemical B, and 26 gallons of chemical C is required to kill a destructive crop insect. Commercial spray X contains one, two and two parts of these chemicals. Spray Y contains only chemical C. Spray Z contains only chemicals A and B in equal amounts. How much of each type of commercial spray is needed to obtain the desired mixture.

(it would mean the world to me if you can also SHOW the three equations created to solve this problem...also, i believe you can work this problem out with the three variable elimination process..thanks!!! COLLEGE SUCKS...haha)

Answer 1

I'm guessing you're dealing with gallons of sprays? If that's the case, then one gallon of X is made up of 20% A, 40% B, and 40% C,
one gallon of Y is 100%C, and
one gallon of Z is made up of 50%A and 50%B.

For chemical A, you have .2x + .5z = 12
For chemical B, you have .4x + .5z = 16
For chemical C, you have .4x + y = 26

Solving for x, use the first two equations simultaneously.
.2x + .5z = 12
.4x + .5z = 16
Subtracting the first line from the second,
.2x = 4
x = 4 / .2 = 20 gallons.

Solving for z, use x = 20 in either equation. (I'll use the first one.)
.2(20) + .5z = 12
4 + .5z = 12
.5z = 8
z = 8 / .5 = 16 gallons.

Solving for y, use x = 20 in the equation for chemical C.
.4x + y = 26
.4(20) + y = 26
8 + y = 26
y = 18 gallons.

You'll need 20 gallons of X, 18 gallons of Y, and 16 gallons of Z.

For quick checks, the (20+18+16 =) 54 gallons of sprays are equal to the (12+16+26=) 54 gallons on chemicals needed. Furthermore, checking the chemical amounts:
Chemical A = .2(20) + .5(16) = 4 + 8 = 12 gallons.
Chemical B = .4(20) + .5(16) = 8 + 8 = 16 gallons.
Cemical C = .4(20) + (18) = 8 + 18 = 26 gallons, the correct amounts of all three.

20 gallons of X, 18 gallons of Y, and 16 gallons of Z is the correct answer.

Answer 2

A must= 12
B must= 16
C must= 26

x(1/5a + 2/5b + 2/5c) + y(c) + z(1/2a + 1/2b) = 12a + 16b + 26c

make a matrix of it - shift y and z right away.

(x)(1/5 2/5 2/5 ) (12)
(z)(1/2 1/2 0 ) = (16)
(y)( 0 0 1 ) (26)

then you solve the matrix :-)

OR....

you can do it without the matrix because they made the problem simpler than it needed to be :-)

you know that you only need enough spray y to flesh out the mixture, set it aside. You have a differentail of 4 gallons between chemical a and chem b, which can't be done with spray z, so set it aside until you have the differential. You will need 20 gallons of spray x to get a 4 gallon differential between chemical A and chemical b (4gal / (2/5 - 1/5) = 20)

current tally: 20 gallons of mix x = 4 gal a, 8 gal b, 8 gal c

next step, add enough mix z to get to 12 and 16 gallons of chems a and b. This is (12-4)/(1/2) = 16gallons

current tally: 20 gallons of mix x + 16 gallons of mix z = 12 gal A, 16 gal B, 8 gal C

finally, add enough of mix y to get the c mixture to the right level - 18 gal

Final answer - 20 gallons mix x
18 gallons mix y
26 gallons mix z

A matrix solution would have been quicker to do, but it's easier to explain this way using text :-)

Answer 3

x = a + 2b + 2c
y = c
z = a + b

the amount of difference between a and b is 4 so that helps in showing how much of x we could use here

4x = 4a + 8b + 8c

thus leaving us with needing 8 gallons of a and b

so we need 8 units of z

thus the remaining amount of c needed is directely input from the remaining spray y

hence what you need is

4x +18y + 8z to make up the required amount of liquid