Once you have expressions for calories from protein, fat, and carbohydrates, add them together and set the sum equal to $180$, the total calories in the bar.

Your equation will have the form $4p + 9f + 4c = 180$. Rearrange this to isolate $f$ on one side. Be careful with signs when you move $4p$ and $4c$ to the other side, and remember you must divide by the coefficient of $f$ at the end.

Think logically: if the bar has more protein or more carbohydrates (which both add calories), should the amount of fat needed to reach 180 total calories go up or down? Use that to decide whether the $(p + c)$ term should be added or subtracted in the expression for $f$.

In Desmos, define the total calories from the macronutrients as an expression: Total(p, f, c) = 4*p + 9*f + 4*c. This corresponds to the equation $4p + 9f + 4c = 180$.

For each answer choice, define a separate function for fat in terms of $p$ and $c$, such as fA(p,c) = 20 + (4/9)*(p + c) for choice A, and similarly for the others. Then, for each one, define a corresponding total, for example TotalA(p,c) = 4*p + 9*fA(p,c) + 4*c.

Create a table in Desmos or use sliders for $p$ and $c$. Plug several different pairs of values for $p$ and $c$ (for example, $(p,c) = (0,0)$, $(5,10)$, $(10,2)$) into each Total expression. The correct answer choice is the one whose total is always $180$ for all tested values of $p$ and $c$.

From the table (using only the Food calories column):

• Protein: $4$ calories per gram
• Fat: $9$ calories per gram
• Carbohydrate: $4$ calories per gram

If the bar has $p$ grams of protein, $f$ grams of fat, and $c$ grams of carbohydrate, then the total calories from each macronutrient are:

• Protein: $4p$
• Fat: $9f$
• Carbohydrate: $4c$

Because the bar has $180$ calories total, we get the equation

We want $f$ in terms of $p$ and $c$, so solve the equation for $f$.

First, move the protein and carbohydrate terms to the other side:

Factor out $4$ on the right side to make the next step easier:

Now divide both sides of $9f = 4(45 - p - c)$ by $9$:

Distribute $\dfrac{4}{9}$:

So the correct expression is

which corresponds to choice B.