A nutritionist is designing a diet plan using two food types, Food A and Food B. The table… — SAT

A nutritionist is designing a diet plan using two food types, Food A and Food B. The table below shows the amount of protein and carbohydrates, in grams, per gram of each food type:

Food Type	Protein (g)	Carbohydrates (g)
Food A	$0.12$	$0.08$
Food B	$0.18$	$0.05$

The nutritionist wants to combine the two foods to obtain a mixture that has a total mass of $250$ grams and contains exactly $36$ grams of protein. How many grams of carbohydrates will be in this mixture?

Answer: 17 grams

Answer

The total amount of carbohydrates in the mixture is 17 grams.

To find the total amount of carbohydrates in the mixture, we first define

x

as the mass in grams of Food A and

y

as the mass in grams of Food B. Since the total mass of the mixture is

250

grams, we write the equation

x + y = 250

. Since the total protein in the mixture must be

36

grams, and Food A and Food B contain

0.12

grams and

0.18

grams of protein per gram respectively, we write the equation

0.12x + 0.18y = 36

. Multiplying the first equation by

12

yields

12x + 12y = 3000

, and multiplying the second equation by

100

yields

12x + 18y = 3600

. Subtracting the first new equation from the second gives

6y = 600

, which simplifies to

y = 100

. Substituting this back into the first equation gives

x + 100 = 250

, so

x = 150

. Using these values, the total amount of carbohydrates is

0.08(150) + 0.05(100) = 12 + 5 = 17

grams.

Step-by-Step Solution

Define variables for the mass of each food type.

Let

x

be the mass, in grams, of Food A, and let

y

be the mass, in grams, of Food B.

To represent the unknown quantities in the system of equations.

Set up the system of linear equations.

x + y = 250

(total mass equation) and

0.12x + 0.18y = 36

(total protein equation).

The mixture has a total mass of

250

grams and must contain exactly

36

grams of protein.

Solve the system of equations for

x

and

y

Multiply the first equation by

12

to get

12x + 12y = 3000

. Multiply the second equation by

100

to get

12x + 18y = 3600

. Subtract the first modified equation from the second to get

6y = 600

, which gives

y = 100

. Substitute

y = 100

back into

x + y = 250

to get

x = 150

To find the specific masses of Food A and Food B required for the mixture.

Calculate the total mass of carbohydrates in the mixture.

Total carbohydrates

= 0.08x + 0.05y = 0.08(150) + 0.05(100) = 12 + 5 = 17

grams.

The question asks for the total grams of carbohydrates in the final mixture.

Key Concept

Systems of Linear Equations

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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