The sum of two numbers is 2 and the sum of their reciprocals is 2.25. Find the numbers.

Let two numbers be x and y.

Given,

Sum of two numbers = 2

⇒ x + y = 2

⇒ x = 2 - y .......(1)

Sum of their reciprocals is 2.25.

⇒ $\frac{1}{x} + \frac{1}{y} = 2.25$

⇒ $\frac{x + y}{xy} = 2.25$ ........(2)

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

$$\begin{aligned}\Rightarrow \frac{2 - y + y}{(2 - y)y} = 2.25 \\[1em] \Rightarrow \frac{2}{2y - y^2} = \frac{225}{100} \\[1em] \Rightarrow 200 = 225(2y - y^2) \\[1em] \Rightarrow 200 = 450y - 225y^2 \\[1em] \Rightarrow 225y^2 - 450y + 200 = 0 \\[1em] \Rightarrow 25(9y^2 - 18y + 8) = 0 \\[1em] \Rightarrow 9y^2 - 18y + 8 = 0 \\[1em] \Rightarrow 9y^2 - 12y - 6y + 8 = 0 \\[1em] \Rightarrow 3y(3y - 4) - 2(3y - 4) = 0 \\[1em] \Rightarrow (3y - 2)(3y - 4) = 0 \\[1em] \Rightarrow (3y - 2)= 0 \text{ or } (3y - 4) = 0 \text{ Using zero product rule } \\[1em] \Rightarrow 3y = 2 \text{ or } 3y = 4 \\[1em] \Rightarrow y = \frac{2}{3} \text{ or } y = \frac{4}{3}. \\[1em]\end{aligned}$$

Hence, the two numbers are $\frac{2}{3},\frac{4}{3}$.