Chapter 8.3, Problem 64E

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 64E

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

  $50$ bags of Seedlings,

  $300$ bags of General potting,

  $50$ 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

  $750$ units of loam

  $450$ 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

  $x+2y+2z=750$ ------(2)

We can write an equation for peat moss as

  $x+y+2z=450$ --------(3)

Therefore,

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

We get

  $\begin{array}{l}\Rightarrow x+2y+2z-\left(x+y+2z\right)=750-450\\ \Rightarrow x+2y+2z-x-y-2z=300\\ \Rightarrow x-x+2y-y+2z-2z=300\\ \Rightarrow y=300\end{array}$

Subtracting equation-(1) with equation-(3)

We get

  $\begin{array}{l}\Rightarrow 2x+y+2z-\left(x+y+2z\right)=500-450\\ \Rightarrow 2x+y+2z-x-y-2z=50\\ \Rightarrow 2x-x+y-y+2z-2z=50\\ \Rightarrow x=50\end{array}$

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

We get

  $\begin{array}{l}2x+y+2z=500\\ \Rightarrow 2\left(50\right)+300+2z=500\\ \Rightarrow 100+300+2z=500\\ \Rightarrow 400+2z=500\\ \Rightarrow 2z=500-400\\ \Rightarrow 2z=100\\ \Rightarrow z=\frac{100}{2}\\ \Rightarrow z=50\end{array}$

Therefore,

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

  $50$ bags of Seedlings,

  $300$ bags of General potting,

  $50$ bags of hardwood potting.