Chapter 8.3, Problem 69E

To determine

To find:Number of bags of potting soil that a company can produce for seedlings, general potting and hardwood plants with given amount of raw materials.

Answer to Problem 69E

Number of bags of potting soil that a company can produce are

  $100$ bags of Seedlings,

  $100$ bags of General potting,

  $100$ bags of hardwood potting.

Explanation of Solution

Given information:

The raw materials used in one bag of each type of plotting soil are given in table.

Available amount of raw materials

  $500$ units of sand

  $500$ units of loam

  $400$ units of peat moss.

Let $\text{'}x\text{'}$ be number of bags for seedlings.

Let $\text{'}y\text{'}$ be number of bags for general potting.

Let $\text{'}z\text{'}$ be number of bags for hardwood potting.

So, from given information

We can write an equation for sand as

  $2x+y+2z=500$ --------(1)

We can equation for loam as

  $x+2y+2z=500$ ------(2)

We can write an equation for peat moss as

  $x+y+2z=400$ --------(3)

Therefore,

Subtracting equation-(2) with equation-(3)

We get

  $\begin{array}{l}\Rightarrow x+2y+2z-\left(x+y+2z\right)=500-400\\ \Rightarrow x+2y+2z-x-y-2z=100\\ \Rightarrow x-x+2y-y+2z-2z=100\\ \Rightarrow y=100\end{array}$

Subtracting equation-(1) with equation-(2)

We get

  $\begin{array}{l}\Rightarrow 2x+y+2z-\left(x+y+2z\right)=500-400\\ \Rightarrow 2x+y+2z-x-y-2z=100\\ \Rightarrow 2x-x+y-y+2z-2z=100\\ \Rightarrow x=100\end{array}$

Substituting values of $x$ and $y$ in equation-(1)

We get

  $\begin{array}{l}2x+y+2z=500\\ \Rightarrow 2\left(100\right)+100+2z=500\\ \Rightarrow 200+100+2z=500\\ \Rightarrow 300+2z=500\\ \Rightarrow 2z=500-300\\ \Rightarrow 2z=200\\ \Rightarrow z=\frac{200}{2}\\ \Rightarrow z=100\end{array}$

Therefore,

Number of bags of potting soil that a company can produce are

  $100$ bags of Seedlings,

  $100$ bags of General potting,

  $100$ bags of hardwood potting.