27 is divided into two parts such that the sum of their reciprocal is $\dfrac{3}{20}$. Find the ratio between the numbers (5 : 4).

Let one number be x and the other number = (27 - x).

The sum of their reciprocal = $\dfrac{3}{20}$

$\Rightarrow \dfrac{1}{x} + \dfrac{1}{27 - x} = \dfrac{3}{20}\\[1em] \Rightarrow \dfrac{27 - x}{x(27 - x)} + \dfrac{x}{x(27 - x)} = \dfrac{3}{20}\\[1em] \Rightarrow \dfrac{27 - x + x}{x(27 - x)} = \dfrac{3}{20}\\[1em] \Rightarrow \dfrac{27}{27x - x^2} = \dfrac{3}{20}\\[1em] \Rightarrow 27 \times 20 = 3 \times (27x - x^2)\\[1em] \Rightarrow 540 = 81x - 3x^2\\[1em] \Rightarrow 81x - 3x^2 - 540 = 0\\[1em] \Rightarrow x^2 - 27x + 180 = 0\\[1em] \Rightarrow x^2 - 12x - 15x + 180 = 0\\[1em] \Rightarrow x(x - 12) - 15(x - 12) = 0\\[1em] \Rightarrow (x - 12)(x - 15) = 0\\[1em] \Rightarrow (x - 12) = 0 \text{ or } (x - 15) = 0\\[1em] \Rightarrow x = 12 \text{ or } x = 15\\[1em]$

If one number = 12, other number = 27 - 12 = 15

If one number = 15, other number = 27 - 15 = 12

Ratio = 12 : 15 = 4 : 5 or 15 : 12 = 5 : 4

Hence, the ratio of two numbers = 5 : 4.