Question

A plant food is to be made from 3 chemicals. The mix must include 60% of the first and second chemicals. The second and third chemicals must be in the ratio of 4:3 by weight. How much of each chemical is needed to make 750kg of the plant food?

Answer 1

The weights of the first, second and third chemicals are 50kg, 300kg, and 400kg respectively.

Explanation:

Let's let the weight of each of the first, second and third chemicals be `a`, `b`, and `c`, and the total weight of the plant food be `d`. The total of each individual component in the mixture must add up to the total weight, so we have our first equation:

`a+b+c=d`

We have also been told that the weight of the first and second chemical represents `60%`, or `0.60` of the total weight, which gives us a second equation:

`a+b=0.60d`

Finally, we are given the ratio of the second and third chemicals as `4:3` leading us to a third equation:

`b/c=4/3`

which we can solve for `c`:

`c=3/4b`

We now have 3 equations and 3 unknowns. Looking at the equations, we notice that the first and second equations differ only by the presence of `c`. Lets take the final equaion for `c` and put it into the first equation to eliminate `c`:

`a+1 3/4b = d`

if we subtract the second equation from this equation we get:

`3/4b=0.4d`

Now we can substitute the total weight of plant food for `d` and solve for `b`:

`b=(4*0.4)/3*750kg=400kg`

Now we can solve for `c` using the rearranged third equation above:

`c=3/4b=3/4*200kg = 300kg`

Finally, lets use the first equation for the total weight rearranged to solve for `a`:

`a=d-b-c=750kg-400kg-300kg = 50kg`

Its always a good idea to check the answers, so lets make sure that we got the percentage of first and second ingredients and ratio of the second and third ingredients correct:

`a+b=50kg+400kg=450kg`

which is `60%` of `750kg` and

`b/c = (400kg)/(300kg) = 4/3` which is the ratio we wanted!