Question

a primitive economy depends on two basic goods, yams and pork. production of 1 bushel of yams requires 1/5 bushels of yams and 1/3 of a pig. to produce 1 pig requires 1/2 bushel of yams. find the amount of each commodity that should be produced to get 300 bushels of yams and 60 pigs. (round to the nearest whole number as needed) to get 300 bushels of yams and 60 pigs requires \\(\square\\) bushels of yams and \\(\square\\) pigs.

Explanation:

Step1: Define variables

Let \( x \) be the number of bushels of yams produced, and \( y \) be the number of pigs produced. We know we want \( x = 300 \) bushels of yams and \( y = 60 \) pigs, but we need to account for the input - output requirements.

The production of 1 bushel of yams requires \( \frac{1}{5} \) bushels of yams (input of yams for yam production) and \( \frac{1}{3} \) of a pig (input of pigs for yam production). The production of 1 pig requires \( \frac{1}{2} \) bushel of yams (input of yams for pig production).

Step2: Set up equations for total yams and total pigs needed

For yams:

The total yams needed (\( Y \)) comes from two parts: the yams needed to produce yams and the yams needed to produce pigs.
The yams needed to produce 300 bushels of yams: Each bushel of yams requires \( \frac{1}{5} \) bushels of yams, so for 300 bushels, it's \( 300\times\frac{1}{5}\) bushels of yams.
The yams needed to produce 60 pigs: Each pig requires \( \frac{1}{2} \) bushel of yams, so for 60 pigs, it's \( 60\times\frac{1}{2}\) bushels of yams.
So the total yams needed \( Y=300 + 300\times\frac{1}{5}+60\times\frac{1}{2}\)
First, calculate \( 300\times\frac{1}{5}=60\) and \( 60\times\frac{1}{2} = 30\)
Then \( Y=300+60 + 30=390\)

For pigs:

The total pigs needed (\( P \)) comes from the pigs needed to produce yams (since pig production does not require pigs as input, only yams).
The pigs needed to produce 300 bushels of yams: Each bushel of yams requires \( \frac{1}{3}\) of a pig, so for 300 bushels, it's \( 300\times\frac{1}{3}\) pigs. And we also need to produce 60 pigs, so the total pigs needed \( P = 300\times\frac{1}{3}+60\)
Calculate \( 300\times\frac{1}{3}=100\), then \( P=100 + 60=160\)

Answer:

390 bushels of yams and 160 pigs.