A nutritionist is blending two protein shake mixes. Mix A contains grams of protein per scoop, and Mix B contains grams per scoop. She wants to prepare exactly scoops of a blend that will average grams of protein per scoop. If Mix A costs \1.20 per scoop, what is the total cost, in dollars, of the -scoop blend?

Solution available with step-by-step explanation

SAT Systems of Two Linear Equations Question 18 | Free SAT Prep | Aniko

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Question 18·Hard·Systems of Two Linear Equations in Two Variables

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A nutritionist is blending two protein shake mixes. Mix A contains $12$ grams of protein per scoop, and Mix B contains $28$ grams per scoop. She wants to prepare exactly $40$ scoops of a blend that will average $20$ grams of protein per scoop.

If Mix A costs $0.50 per scoop and Mix B costs $1.20 per scoop, what is the total cost, in dollars, of the $40$ -scoop blend?

For mixture problems with an average and a total number of items, first convert the average into a total amount (average × number of items). Then define variables for how much of each component you use and write two equations: one for the total count (like $a + b = \text{total scoops}$ ) and one for the total quantity being averaged (like protein, cost, or volume). Solve this system using substitution or elimination, and finally plug the resulting amounts into the cost expression. Keeping the steps in this order—total, equations, solve, then cost—helps prevent common mistakes and saves time.

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Turn the average into a total

You know the blend will have 40 scoops and should average 20 grams of protein per scoop. How many total grams of protein does that mean the entire blend must contain?

Introduce variables and write equations

Let one variable represent the number of scoops of Mix A and another for Mix B. You know their sum is 40 scoops, and you also know how many grams of protein each scoop contributes. How can you write an equation for the total grams of protein?

Solve the system, then find the cost

Use substitution or elimination to solve the two equations for the numbers of scoops of each mix. Once you know how many scoops of each mix are used, how can you combine the costs per scoop to get the total cost?

Desmos Guide

Graph the system for scoop counts

Use $x$ for the number of scoops of Mix A and $y$ for the number of scoops of Mix B. In Desmos, enter the two equations as

y = 40 - x (from $x + y = 40$ )
y = (800 - 12x) / 28 (from $12x + 28y = 800$ )

Look for the intersection point of the two lines; its coordinates $(x, y)$ give the numbers of scoops of each mix.

Use the intersection to compute the total cost

Once you read off the intersection values for $x$ and $y$ , type a new expression in Desmos: 0.5 * x + 1.2 * y, replacing $x$ and $y$ with the intersection coordinates you found. The value that Desmos outputs for this expression is the total cost of the 40-scoop blend in dollars.

Step-by-step Explanation

Translate the average into a total amount of protein

The blend has 40 scoops and should average 20 grams of protein per scoop.

So the total grams of protein in the blend must be

40 \times 20 = 800 \text{ grams}.

Set up equations for the numbers of scoops

Let $a$ be the number of scoops of Mix A and $b$ be the number of scoops of Mix B.

Total scoops:
$a + b = 40$ .
Total protein: each scoop of A has 12 g, each scoop of B has 28 g, and together they must add to 800 g:

12a + 28b = 800.

Solve the system of equations for a and b

Use substitution from the first equation.

From $a + b = 40$ :

a = 40 - b.

Substitute into the protein equation:

12(40 - b) + 28b = 800.

Distribute and simplify:

\begin{aligned} 480 - 12b + 28b &= 800 \\ 480 + 16b &= 800. \end{aligned}

Solve for $b$ :

\begin{aligned} 16b &= 800 - 480 \\ 16b &= 320 \\ b &= 20. \end{aligned}

Then from $a + b = 40$ :

a + 20 = 40 \Rightarrow a = 20.

So the blend uses 20 scoops of each mix.

Compute the total cost of the blend

Mix A costs $0.50 per scoop and Mix B costs $1.20 per scoop.

Total cost is

0.50(20) + 1.20(20).

Calculate each part:

0.50(20) = 10, \quad 1.20(20) = 24.

Add them:

10 + 24 = 34.

So, the total cost of the 40-scoop blend is $34, which corresponds to answer choice C.

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation