To find the width of the pathway, we can set up an equation using the given information:

1. The garden has dimensions of 15 meters by 18 meters.
2. The total area including the pathway is 360 square meters.

Let $x$ be the width of the pathway. The dimensions of the entire area, including the garden and the pathway, will be $(15 + 2x) \times (18 + 2x)$.

The equation for the total area is: $(15 + 2x)(18 + 2x) = 360$

Expanding this equation: $270 + 30x + 36x + 4x^2 = 360$

Combine like terms: $4x^2 + 66x + 270 = 360$

Subtract 360 from both sides: $4x^2 + 66x - 90 = 0$

Simplify by dividing the whole equation by 2: $2x^2 + 33x - 45 = 0$

Now use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where $a = 2$, $b = 33$, and $c = -45$.

Calculate the discriminant: $b^2 - 4ac = 33^2 - 4 \times 2 \times (-45) = 1089 + 360 = 1449$

Calculate $x$: $x = \frac{-33 \pm \sqrt{1449}}{4}$

Since we need a positive width, we take the positive solution. Approximating the square root gives: $x = \frac{-33 + \sqrt{1449}}{4}$

This will give you the numeric value for the width of the pathway.