Mixture problem

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A mixture of 12 liters of chemical A, 16 liters of chemical B, and 26 liters of chemical C is required to kill a destructive crop insect. Commercial spray X contains 1, 2, and 2 parts respectively, of these chemicals, Commercial spray Y contains only chemical c. Commercial spray Z contains only chemicals A and B in equal amounts. How much of each type of commercial spray is needed to get the desired mixture?
Would I set it up so that a+2b+2c=12 a+b=16 and c=26? I really don't know where to go. Can you help me?

thanks in advance
 
I would approach it as:
If you use x units of X you get
ax+2bx+2cx
If you use y units of Y you get
cy
If you use z units of Z you get
az+bz

When you add them up (down?) you get
x+z units of a = 12
2x+z units of b = 16
2x+y units of c = 26
Solve it.
 
Hello, americo74!

Your equations are incorrect.
. . But it's easy to do that ... I know!

A mixture of 12 liters of chemical A, 16 liters of chemical B, and 26 liters of chemical C is required.
Commercial spray X contains 1, 2, and 2 parts respectively, of these chemicals,
Commercial spray Y contains only chemical C.
Commercial spray Z contains only chemicals A and B in equal amounts.
Click to expand...

Let \(\displaystyle x\) = liters of spray X.
Let \(\displaystyle y\) = liters of spray Y.
Let \(\displaystyle x\) = liters of spray Z.

Here' s the baby-talk I used . . .
. . Spray X used: \(\displaystyle x\) liters of A, \(\displaystyle 2x\) liters of B, \(\displaystyle 2x\) liters of C.
. . Spray Y used: \(\displaystyle y\) units of C.
. . Spray Z used: \(\displaystyle z\) liters of A, \(\displaystyle z\) liters of B.

I put them into a chart:
Code: 
        | A |  B |  C |
     ---+---+----+----+
      X | x | 2x | 2c |
     ---+---+----+----+
      Y |   |    |  y |
     ---+---+----+----+
      Z | z |  z |    |
     ---+---+----+----+
         12   16   26
Now read <u>down</u> the columns . . .
. . There are our equations!

. . . . . . \(\displaystyle x\;\;\;\;\;+\,z\;=\;12\)
. . . . . \(\displaystyle 2x\;\;\;\;\;+\,z\;=\;16\)
. . . . . \(\displaystyle 2x\,+\,y\;\;\;\;\,\,\,=\;26\)


[Edit: Beat me again, Gene!\\]
 
americo74 said:
A mixture of 12 liters of chemical A, 16 liters of chemical B, and 26 liters of chemical C is required to kill a destructive crop insect. Commercial spray X contains 1, 2, and 2 parts respectively, of these chemicals, Commercial spray Y contains only chemical c. Commercial spray Z contains only chemicals A and B in equal amounts. How much of each type of commercial spray is needed to get the desired mixture?
Would I set it up so that a+2b+2c=12 a+b=16 and c=26? I really don't know where to go. Can you help me?

thanks in advance
Click to expand...

Start by forgetting about spray Y.
Chemical A needs x from X and z from Z: x + z = 12 [1]

Chemical B needs 2x from X and z from Z: 2x + z = 16 [2]

[1][2]: x = 4, z = 8

Now bring in Y:
Chemical C: 2x still required from X, so: y = 26 - 2x = 26 - 8 = 18

X :Y : Z
4 : 0 : 8 = 12
8 : 0 : 8 = 16
8 :18: 0 = 26
 
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