Expert: Sherry Wallin - 10/5/2010

Question
Dudley is a health-conscious student whose diet consists entirely of fried chicken, pizza and chocolate milkshakes. He has done some research on the internet and
found the following information:
• 1 serve of fried chicken contains 20 g of fat, 45 g of protein, and no carbohydrate.
• 1 slice of pizza contains 10 g of fat, 15 g of protein and 30 g of carbohydrate.
• 1 chocolate milkshake contains 50 g of fat, 30 g of protein and 80 g of carbohydrate.
• For a balanced diet, he should consume 90 g of fat, 60 g of protein and 320 g
of carbohydrate each day.

Find the number of serves of fried chicken, the number of slices of pizza and the number of milkshakes that Dudley should consume each day in order to maintain a balanced diet;

OR show that it is not possible for him to maintain a balanced diet if he continues to consume nothing other than fried chicken, pizza and milkshakes.

thanks :)

Answer
Julie~

It is impossible to get the proper nutrition, i.e., the balanced diet eating any combination of fried chicken, pizza, and chocolate milkshakes.

Let f = # of servings of fried chicken
let p = # of servings of pizza
let c = # of servings of chocolate milkshakes

the total number of fat grams must add up to 90:
20f + 10p + 50c = 90 => 2f + p + 5c = 9  divided each term by 10

and the total number of protein grams must add up to 60:
45f + 15p = 60 => 3f + p      = 4  divided each term by 15

and the total number of carbohydrate grams must add up to 320:
50f + 30p + 80c = 320 => 5f + 3p + 8c = 32

This your system of equations that represents the data given. Solve the system. I like to number the equations to make it easier to follow.

(eqn1)  2f +  p + 5c   = 9
(eqn2)  3f +  p           = 4
(eqn3)  5f + 3p + 8c  = 32

eqn1 - eqn2: call this new equation eqn4
2f-3f + p-p + 5c-0 = 9-4 => -f + 5c = 5

neg 3 times eqn1 added to eqn3: -3eqn1 + eqn3 call this equation eqn5
-3(2f) + 5f + -3p + 3p + -3(5c) + 8c = -3(9) + 32 =>
-f + -7c = 5

eqn4 - eqn3:
-f -(-f) + 5c -(-7c) = 5-5 =>
12c = 0 => c = 0 => no fried chicken will be allowed in his diet

Substitute c = 0 into eqn 4:
-f + 5(0) = 5 =>
-f = 5 =>
f = -5 ** you can't have -5 servings of anything, let alone pizza!

Therefore, it is impossible to get a balanced diet with the fried chicken, pizza, and chocolate milkshakes.

Math Prof