Expert: Paul Klarreich - 10/7/2009

Question
Hi there Paul, I am looking for some clarification on a certain word problem shown below. The course I am in is Applications of Linear Models and right now we are studying Elementary Matrices. At the end of the question it asks me to solve using matrix methods. You may not be able to help me with this part but I want to be sure that I am using the right algebraic equations to get started. Thanks in advance for your help.

A fertilizer company produces two lawn fertilizers, regular and deluxe. Each 20 kilogram bag of fertilizer contains 40%active ingredients and 60%inert substances. Each 20 kilogram bag of regular fertilizer contains 20% active ingredients and 80% inert substances. The company anticipates having an inventory of 30,000 kilograms of active ingredients and 60,000 kilograms of inert substances, and wishes to use the entire inventory in production of the two fertilizers. Given the information provided above, how many bags of each type of fertilizer should be produced by the company? Use matrix methods to answer this question.

These are the equations I have come up with so far. But I am not sure if they are right.

A=active and I=inert

8(A)+ 12(I)= 20
4(A)+ 16(I)= 20 (these are the percentages converted into kgs/bag)
30 000(A)+ 60 000(I)= 90 000

Answer
Questioner: Dane
Country: Canada
Category: Advanced Math
Private: No
Subject: Word Problem
Question: Hi there Paul, I am looking for some clarification on a certain word problem shown below. The course I am in is Applications of Linear Models and right now we are studying Elementary Matrices. At the end of the question it asks me to solve using matrix methods. You may not be able to help me with this part but I want to be sure that I am using the right algebraic equations to get started. Thanks in advance for your help.

A fertilizer company produces two lawn fertilizers, regular and deluxe. Each 20 kilogram bag of fertilizer contains 40%active ingredients and 60%inert substances. Each 20 kilogram bag of regular fertilizer contains 20% active ingredients and 80% inert substances. The company anticipates having an inventory of 30,000 kilograms of active ingredients and 60,000 kilograms of inert substances, and wishes to use the entire inventory in production of the two fertilizers. Given the information provided above, how many bags of each type of fertilizer should be produced by the company? Use matrix methods to answer this question.

These are the equations I have come up with so far. But I am not sure if they are right.

A=active and I=inert

8(A)+ 12(I)= 20
4(A)+ 16(I)= 20 (these are the percentages converted into kgs/bag)
30 000(A)+ 60 000(I)= 90 000
========================================
Hi, Dane,

I don't like this part:

You wrote:

A fertilizer company produces two lawn fertilizers, regular and deluxe.
>> OK, so far.

Each 20 kilogram bag of (DELUXE?) fertilizer contains 40%active ingredients and 60%inert substances.
>> Not so OK.  I like:

Deluxe 'fert' contains 40% active.

Each 20 kilogram bag of regular fertilizer contains 20% active ingredients and 80% inert substances.

>> Also not OK.  I like:

Regular 'fert' contins 20% active.

Then you wrote:

A=active and I=inert

I REALLY REALLY don't like that.  Instead:

Let  D = number of kg of deluxe  produced.
Let  R = number of kg of regular produced.

Then you observe:

We will use  30000 kg of active stuff.
We will make 90000 kg of product.

.40 D + .20 R = 30000   << active ingredient.
   D + R = 90000   << product.

I think them's your equations.

In matrix form, I think it looks like:

( 0.4  0.2 )( D ) = (30000)
(   1    1 )( R ) = (90000)

This is really one of your ninth-grade mixture problems.  (Well, in the New York State syllabus, it's ninth-grade algebra.)