We are given that $x$ is directly proportional to $y$ and inversely proportional to $z$. Also, $x = 15$ when $y = 10$ and $z = 4$. We need to find the equation connecting $x$, $y$, and $z$.

1. Problem Description

We are given that $x$ is directly proportional to $y$ and inversely proportional to $z$. Also, $x = 15$ when $y = 10$ and $z = 4$. We need to find the equation connecting $x$, $y$, and $z$.

2. Solution Steps

Since $x$ is directly proportional to $y$ and inversely proportional to $z$, we can write the relationship as:

$x = k \frac{y}{z}$

where $k$ is the constant of proportionality.

We are given that $x = 15$ when $y = 10$ and $z = 4$. Substituting these values into the equation, we get:

$15 = k \frac{10}{4}$

Multiplying both sides by 4, we have:

$15 \times 4 = k \times 10$

$60 = 10k$

Dividing both sides by 10, we get:

$k = \frac{60}{10} = 6$

Therefore, the equation connecting $x$, $y$, and $z$ is:

$x = 6 \frac{y}{z} = \frac{6y}{z}$

3. Final Answer

The equation connecting $x$, $y$, and $z$ is $x = \frac{6y}{z}$.

Therefore, the answer is A.