Let $t$ be the time (in minutes) to make 120 gadgets. How can you write a proportion that compares $15$ gadgets in $4$ minutes to $120$ gadgets in $t$ minutes?

Once you have a proportion like $\frac{15}{4} = \frac{120}{t}$, what equation do you get when you cross-multiply, and how do you solve it for $t$?

In Desmos, type the expression 480/15 (or equivalently 120/(15/4) or (120*4)/15) and look at the numerical output; that value is the number of minutes the factory needs to produce 120 gadgets.

The problem says the factory produces 15 gadgets every 4 minutes.

So the rate can be written as a ratio:

This means in 4 minutes, the factory makes 15 gadgets.

Let $t$ be the number of minutes needed to produce 120 gadgets at the same rate.

Set up a proportion comparing gadgets to time:

Here, both fractions are "gadgets per minute" (gadgets on top, minutes on bottom).

Now solve the equation

by cross-multiplying:

So

Now divide both sides by 15 to isolate $t$:

Now simplify $\frac{480}{15}$:

• First divide numerator and denominator by 5: $\frac{480}{15} = \frac{96}{3}$.
• Then divide: $\frac{96}{3} = 32$.

So $t = 32$. This means it will take 32 minutes for the factory to produce 120 gadgets at this rate.