7 m = 5 n + p The given equation relates the positive numbers m ,

Choice B is correct. It's given that the equation $7m=5\left(n+p\right)$ relates the positive numbers $m$, $n$, and $p$. Dividing both sides of the given equation by $5$ yields $\frac{7m}{5}=n+p$. Subtracting $p$ from both sides of this equation yields $\frac{7m}{5}-p=n$, or $n=\frac{7m}{5}-p$. It follows that the equation $n=\frac{7m}{5}-p$ correctly gives $n$ in terms of $m$ and $p$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.