Define variables

Let $x$, $y$, and $z$ be the time taken (in hours) for the first, second, and third pipes respectively.

$x, y, z$

Establish time relations

From the problem, the second pipe fills the tank 5 hours faster than the first ($x = y + 5$) and 4 hours slower than the third ($y = z + 4$).

$x = y + 5, y = z + 4$

Formulate combined rate equation

The first and second pipes working together fill the tank in the same time as the third pipe alone.

$\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$

Substitute and solve

Substitute $y = x - 5$ and $z = y - 4 = x - 9$ into the rate equation: $1/x + 1/(x-5) = 1/(x-9)$. Solving for $x$ gives $x=15$.

$\frac{1}{15} + \frac{1}{10} = \frac{1}{6}$