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# Quinoa Pasta 2

Alignments to Content Standards: A-REI.C.6

A type of pasta is made of a blend of quinoa and corn. How much of the pasta is quinoa and how much is corn? The pasta company is not disclosing the percentage of each ingredient in the blend. However, quinoa contains a higher percentage of protein than corn. At the company’s website, they give some information about the protein content of different foods and their own pasta blend, see http://www.quinoa.net/199.html. Use the protein content of each ingredient to find out how much quinoa and how much corn is in one serving of the pasta.

## IM Commentary

This is a full-blown modeling problem (SMP 4). Students must come up with the idea on their own that they can set up two equations in two unknowns to solve the problem, and they must then read, understand and extract information from the nutrition labels to set up the equations in the system (SMP 1). The algebra needed to solve the system is fairly simple substitution; the main difficulty lies in the modeling aspect.

This task is a variant of 8.EE Quinoa Pasta 1, where all the relevant information is given as part of the task statements and the students are asked to set up a system of equations. Another variant of this task is given in A-REI.6 Quinoa Pasta 3, which is a more open-ended modeling task. It poses the question by showing a photo of the box, but the students have to formulate a plan to answer it.

It is worth spending some time discussing reasonable levels of numerical precision. While the percentage of quinoa which is protein is given to three digits of accuracy, other values in the problem (e.g., the number of grams of protein per serving) are given only to the nearest gram. So while the computations shown in the solution are carried out using 3 decimal places, the answer should only be reported as whole numbers.

This task could be used as an instructor led activity or as a group activity where students cooperate to formulate a plan of how to solve the problem and to set up the equations required to solve the problem.

## Solution

The information on the website states in the table titled “Quinoa’s Food Value” that quinoa contains 16.2% protein and corn contains 3.5% of protein. The table at the bottom of the page titled “Gluten Free Pasta Nutrition Facts” lists that the pasta blend contains 4 grams of protein in a serving of size 57 grams. This is enough information to solve the problem with a system of two equations in two unknowns.

If we let $q$ be the amount of quinoa, in grams, in one serving of pasta and $c$ be the amount of corn, in grams, in one serving of pasta we have $q + c = 57$. We also know that 16.2% of quinoa is protein and 3.5% of corn is protein and one serving of pasta contains 4 grams of protein. We can summarize this information in the equation $0.162q + 0.035c = 4.$ Therefore, we have the following system of equations:

\begin{align} q + c &= 57 \\ 0.162q + 0.035c &= 4. \end{align}

We can solve this system using the method of substitution or the method of elimination. Using the method of substitution, we solve the first equation for $q$:

$$q = 57 − c.$$

We substitute this for $q$ in the second equation and solve for $c$ :

\begin{align} 0.162q + 0.035c &= 4 \\ 0.162(57 - c) + 0.035c &= 4 \\ 9.234 - 0.162c + 0.035c &= 4 \\ -0.127c &= 4 - 9.234 \\ -0.127c &= -5.234 \\ c &= 41.213. \end{align}

So we have $c=41$ and $q+c=57$, therefore $q=16$. Out of the 57 grams of pasta in one serving, 41 grams are corn and 16 grams are quinoa. In other words, about 72% of the pasta blend is corn and 28% is quinoa.

Note: In our solution to the problem, we are making the assumption that the protein content of the corn and the quinoa used to make the pasta is the same as the protein content listed in the table. This may or may not be the case. For example, Wikipedia lists the protein content of quinoa as 14% and of corn as 3.22%. Using these numbers, our solution would be different: 35% quinoa, 65% corn.