Question 203031: 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
chemical A and B in equal amounts. How much of each type of commercial spray is needed to get the
desired mixture?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! -----
product X contains 1 2 2 parts A B C respectively.
product Y contains 0 0 1 parts A B C respectively.
product Z contains 1 1 0 parts A B C respectively.
-----
you can only buy whole products, so:
let f = number of product X
let g = number of product Y
let h = number of product Z
-----
you will want to buy
f*X + g*Y + h*Z
-----
f*X = f*(1A + 2B + 2C)
g*Y = g*(0A + 0B + 1C)
h*Z = h*(1A + 1B + 0C)
-----
since the total liters of chemical A = 12, then:
f*1 + g*0 + h*1 = 12 which can be written as 1f + 0g + 1h = 12
since the total liters of chemical B = 16, then:
f*2 + g*0 + h*1 = 16 which can be written as 2f + 0g + 1h = 16
since the total liters of chemical C = 28, then:
f*2 + g*1 + h*0 = 26 which can be written as 2f + 1g + 0h = 26
-----
you have a series of equations that need to be solved simultaneously to get the same value for f, g, and h in each.
-----
the equations are:
1f + 0g + 1h = 12
2f + 0g + 1h = 16
2f + 1g + 0h = 26
-----
if You subtract the second equation from the first, You get:
f = 4
-----
if You substitute f = 4 in either the first or second equation, You get:
h = 8
You will get the same answer using either one.
-----
if You substitute f = 4 in the third equation, You get:
g = 18
-----
your answer should be:
f = 4
g = 18
h = 8
-----
To prove this is true, substitute those values in the following product equation.
f*X + g*Y + h*Z
which represents the number of each product you need to buy.
f*X = 4 * (1A + 2B + 2C) = 4A + 8B + 8C
g*Y = 18 * (0A + 0B + 1C) = 18C
h*Z = 8* (1A + 1B + 0C) = 8A + 8B.
-----
the total liters of A is 4 + 8 = 12
the total liters of B is 8 + 8 = 16
the total liter of C is 8 + 18 = 26
-----
since the total liters of chemicals A, B, C check out ok, then You need to buy:
4 packages of product X.
18 packages of product Y.
8 packages of product Z.
under the assumption that the parts of each chemical listed in each product was the liters of each chemical in that product, and not just the ratio, i.e. product Y ratio was 0 0 1 which is interpreted to mean 0 liters of chemical A and 0 liters of chemical B and 1 liter of chemical C are contained in 1 package of product Y for a total of 1 liter.
Similarly product Z is interpreted to contain 1 liter of chemical A and 1 liter of chemical B for a total of 2 liters.
-----
|
|
|
