Every gram of wheat provides 0.15 g of proteins and 0.24 g of carbohydrates. The corresponding values of proteins and carbohydrate of rice are 0.05g and 0.48g respectively. Wheat costs Rs.40 per kg and rice costs Rs.60 per kg. The minimum daily requirements of proteins and carbohydrates for an average child are 50g and 200 g respectively. Which of the following options describes the LPP to minimize the total cost of wheat and rice that provide minimum daily requirements of proteins and carbohydrates?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → minimize \( z=\displaystyle\frac{40x}{1000}+\displaystyle\frac{60y}{1000} \) subjected to constraints \( 0.15x+0.05y\ge 50 \), \( 0.24x+0.48y\ge 200,x,y\ge 0 \)
Let the child be given x g of wheat and y g of rice. The nutrient data are per gram, so x and y are in grams.
Cost: Rs. 40 per kg is Rs. \( \frac{40}{1000} \) per gram, and Rs. 60 per kg is Rs. \( \frac{60}{1000} \) per gram:
\( \displaystyle z=\frac{40x}{1000}+\frac{60y}{1000} \)
Constraints: the requirements are minimum amounts, so the intake must be at least 50 g of proteins and 200 g of carbohydrates:
\( \displaystyle 0.15x+0.05y\ge 50 \)
\( \displaystyle 0.24x+0.48y\ge 200 \)
\( \displaystyle x,y\ge 0 \)
This is Option (3).
Why the other options are incorrect
Option (1) minimize \( z=40x+60y \) subjected to constraints \( 0.15x+0.05y\ge 60 \), \( 0.24x+0.48y\ge 400,x,y\ge 0 \) — Incorrect. The cost 40x + 60y uses prices per kg with quantities in grams, and the right-hand sides 60 and 400 are not the given requirements of 50 g and 200 g.
Option (2) minimize \( z=40x+60y \) subjected to constraints \( 0.15x+0.05y\le 50 \), \( 0.24x+0.48y\le 200,x,y\ge 0 \) — Incorrect. The cost uses prices per kg with quantities in grams, and “≤” would cap the nutrients instead of guaranteeing the minimum requirement.
Option (4) minimize \( z=\displaystyle\frac{40x}{1000}+\displaystyle\frac{60y}{1000} \) subjected to constraints \( 0.15x+0.05y\le 50 \), \( 0.24x+0.48y\le 200,x,y\ge 0 \) — Incorrect. The cost is right, but “≤” would cap the nutrients instead of guaranteeing the minimum requirement.