If $\dfrac{1}{4}$ inch represents 3 feet, how many groups of $\dfrac{1}{4}$ inch are in 1 inch, and how does that change the number of feet?

Once you know how many feet 1 inch represents, multiply that by $x$ to find how many feet $x$ inches represents.

In Desmos, type 3 / (1/4) to compute how many feet correspond to 1 inch on the blueprint. Note the numerical result; this is the number of feet per inch.

Now type (3 / (1/4)) * x in Desmos. The coefficient of $x$ in this expression is the number of feet represented by each inch, so the whole expression gives the actual hallway length in feet in terms of $x$.

The blueprint uses the scale: $\dfrac{1}{4}$ inch on the drawing represents $3$ feet in real life.

So the ratio is

We want the number of feet per 1 inch on the blueprint.

If $\dfrac{1}{4}$ inch $= 3$ feet, then 1 inch is 4 times as long as $\dfrac{1}{4}$ inch, so it represents 4 times as many feet:

So, 1 inch on the blueprint represents 12 feet in real life.

Now the hallway is $x$ inches long on the blueprint.

Since each 1 inch represents 12 feet, $x$ inches will represent $x$ groups of 12 feet:

So, the actual length of the hallway, in feet, is $12x$.