Among the failed widgets, what fraction is blue? Write that fraction using $x$ in the numerator and denominator.

Once you have the fraction for "blue among failed," set it equal to $2/5$ and solve the resulting equation for $x$.

In Desmos, type y = x/(30 + x) on one line and y = 2/5 on another line to graph both the probability expression and the constant $2/5$.

Use the point-of-intersection feature (tap/click where the curves cross, or use the menu) to find where the graphs of y = x/(30 + x) and y = 2/5 intersect. The x-coordinate of that intersection is the value of $x$ that satisfies the given probability condition.

The probability is about failed widgets only, so ignore the "Passed inspection" row.

From the "Failed inspection" row:

• Red: 30
• Blue: $x$

So the total number of failed widgets is $30 + x$.

The problem says: "If one of the failed widgets is selected at random, the probability that the widget is blue is $\dfrac{2}{5}$."

Among failed widgets:

• Number of blue failed widgets $= x$
• Total failed widgets $= 30 + x$

So the probability that a failed widget is blue is

We are told this equals $\dfrac{2}{5}$, so we set up the equation

We solve the equation

Cross-multiply:

Distribute on the right side:

Subtract $2x$ from both sides:

Divide both sides by 3:

So, the value of $x$ is 20.