Question

a primitive economy depends on two basic goods, yams and pork. production of 1 bushel of yams requires 1/7 bushels of yams and 1/2 of a pig. to produce 1 pig requires 1/3 bushel of yams. find the amount of each commodity that should be produced to get 300 bushels of yams and 50 pigs.
to get 300 bushels of yams and 50 pigs requires \\(\square\\) bushels of yams and \\(\square\\) pigs.
(round to the nearest whole number as needed)

Explanation:

Let \( x \) be the number of bushels of yams and \( y \) be the number of pigs needed.

Step 1: Set up equations for yams and pigs

• For yams: The production of 300 bushels of yams requires \( \frac{1}{7} \) bushel of yams per bushel of yams produced, and the production of 50 pigs requires \( \frac{1}{3} \) bushel of yams per pig produced. So the total yams needed for yams production is \( \frac{1}{7} \times 300 \), and for pigs production is \( \frac{1}{3} \times 50 \). Thus, the equation for yams is:

\( x=\frac{1}{7}\times300+\frac{1}{3}\times50 \)

• For pigs: The production of 300 bushels of yams requires \( \frac{1}{2} \) pig per bushel of yams produced, and the production of 50 pigs requires 1 pig per pig produced (since we just need 50 pigs). So the equation for pigs is:

\( y = \frac{1}{2}\times300+50 \)

Step 2: Calculate yams needed

First, calculate \( \frac{1}{7}\times300=\frac{300}{7}\approx42.86 \) and \( \frac{1}{3}\times50=\frac{50}{3}\approx16.67 \). Then add them together:
\( x=\frac{300}{7}+\frac{50}{3}=\frac{900 + 350}{21}=\frac{1250}{21}\approx59.52\approx60 \) (rounded to nearest whole number)

Step 3: Calculate pigs needed

Calculate \( \frac{1}{2}\times300 = 150 \), then add 50:
\( y=150 + 50=200 \)

Answer:

60 bushels of yams and 200 pigs.