The sum of two natural numbers is 5 and the sum of their reciprocals is $\dfrac{5}{6}$, the numbers are :

1. 2 and 5

2. 4 and 2

3. 2 and 3

4. 3 and 4

Let two natural numbers be x and y.

Given,

Sum = 5

⇒ x + y = 5

⇒ x = 5 - y ……..(1)

Sum of reciprocals = $\dfrac{5}{6}$

⇒ $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{5}{6}$ ………..(2)

Substituting value of x from equation (1) in (2) :

$\Rightarrow \dfrac{1}{5 - y} + \dfrac{1}{y} = \dfrac{5}{6} \\[1em] \Rightarrow \dfrac{y + 5 - y}{y(5 - y)} = \dfrac{5}{6} \\[1em] \Rightarrow \dfrac{5}{5y - y^2} = \dfrac{5}{6} \\[1em] \Rightarrow 5y - y^2 = \dfrac{6 \times 5}{5} \\[1em] \Rightarrow 5y - y^2 = 6 \\[1em] \Rightarrow y^2 - 5y + 6 = 0 \\[1em] \Rightarrow y^2 - 2y - 3y + 6 = 0 \\[1em] \Rightarrow y(y - 2) - 3(y - 2) = 0 \\[1em] \Rightarrow (y - 3)(y - 2) = 0 \\[1em] \Rightarrow y - 3 = 0 \text{ or } y - 2 = 0 \\[1em] \Rightarrow y = 3 \text{ or } y = 2.$

If y = 3,

x = 5 - y = 5 - 3 = 2.

If y = 2,

x = 5 - y = 5 - 2 = 3.

∴ Numbers are 2 and 3.

Hence, Option 3 is the correct option.