Step 1: Recall the max and min intensity formulas. In a double slit pattern the brightest and darkest spots are \[ I_{max} = (\sqrt{I_1} + \sqrt{I_2})^2,\quad I_{min} = (\sqrt{I_1} - \sqrt{I_2})^2 \]
Step 2: Use the given ratio. We are told $\dfrac{I_{max}}{I_{min}} = \dfrac{25}{9}$. Take the square root of both sides. \[ \frac{\sqrt{I_1} + \sqrt{I_2}}{\sqrt{I_1} - \sqrt{I_2}} = \frac{5}{3} \] Step 3: Use simple letters. Let $a = \sqrt{I_1}$ and $b = \sqrt{I_2}$. \[ \frac{a + b}{a - b} = \frac{5}{3} \] Step 4: Cross multiply. \[ 3(a + b) = 5(a - b) \] \[ 3a + 3b = 5a - 5b \] Step 5: Solve for the ratio of a to b. \[ 8b = 2a \quad\Rightarrow\quad a = 4b \] Step 6: Get the intensity ratio. Square the relation since $I = (\text{root})^2$. \[ \frac{I_1}{I_2} = \frac{a^2}{b^2} = 16:1 \] \[ \boxed{16:1} \]