Given: $\frac{x}{7}+\frac{w}{28}=1$

Multiply both sides of the equation by $28$ to get: $4x+w=28$

Since $x$ and $w$ must be positive integers, let's start listing the possible solutions...
$x = 1$ and $w = 24$
$x = 2$ and $w = 20$
$x = 3$ and $w = 16$
$x = 4$ and $w = 12$
$x = 5$ and $w = 8$
$x = 6$ and $w = 4$
$x = 7$ and $w = 0$ doesn't work because w is not positive in this case

There are 6 solutions in which x and w are positive integers.

Answer: B