If x varies directly as y and inversely as z, and x = 16 when y = 40 and z = 3, then find x when y = 50 and z = 6.

The correct option is 10.

Step 1: Set up the variation equation
We are given that $x$ varies directly as $y$ and inversely as $z$. This relationship can be expressed mathematically as:

$x=k·\frac{y}{z}$

where $k$ is the constant of variation.

Step 2: Find the constant of variation ($k$)
We are given the initial conditions: $x=16$ when $y=40$ and $z=3$. Substitute these values into the variation equation:

$16=k·\frac{40}{3}$

To solve for $k$, multiply both sides by $\frac{3}{40}$:

$k=16·\frac{3}{40}$

Simplify the fraction:

$k=\frac{16·3}{40}=\frac{48}{40}=\frac{6}{5}$

Step 3: Calculate the new value of $x$
Now we need to find $x$ when $y=50$ and $z=6$, using the value of $k=\frac{6}{5}$:

$x=\frac{6}{5}·\frac{50}{6}$

Simplify the expression by canceling out the common terms:

$x=\frac{6·50}{5·6}=\frac{50}{5}=10$

Thus, the value of $x$ is 10.