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?

1 Answer
Feb 22, 2016

Answer:

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!