SOLUTION: A plant food is to be made from three chemicals. The combined amounts of the first two chemicals must be 80% of the resulting mix. The second and third chemical must be in the rati

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Question 708648: A plant food is to be made from three chemicals. The combined amounts of the first two chemicals must be 80% of the resulting mix. The second and third chemical must be in the ratio 4 to 3 by weight. How much of each chemical is needed to make 700 kg of the plant food?
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Use a, b, and c as the quantities each chemical.
These initial equations can be made from the problem description:
The total was to be 700 pounds, so a%2Bb%2Bc=700.
The 80% requirement gives, %28a%2Bb%29%2F700=0.8
The ratio information indicates, b%2Fc=4%2F3.

Rework the last two equations into these:
a%2Bb=560, and b-%284%2F3%29c=0.

The system of equations to work with then is this one:
a%2Bb%2Bc=700
a%2Bb=560
b-%284%2F3%29c=0

Matrix operations is the way to continue here, at least partly.
That system can give us this matrix:


I briefly list the row operations but not here show the work.
R%5B1%5D-R%5B2%5D=+R%5B2%5D, immediately giving c=140.
R%5B3%5D%2B%284%2F3%29R%5B1%5D=R%5B3%5D, giving b=186 2/3, b=186%262%2F3.

Now having b and c, the 700 summation equation can be used and b and c simply subsituted to find a (not continuing further with matrix operations).
result for a: a=373%262%2F3

ANSWER: a=373%261%2F3, b=186%262%2F3, c=140, all in pounds.