The numbers are 2 and 1/2Step-by-step explanation:Given :The sum of a number and it's reciprocal = 5/2To FIND : NumbersLet the 1st number be x and the other be 1/xAccording to Question\begin{gathered}x + \frac{1}{x} = \frac{5}{2} \\ \\ \END{gathered}x+x1=25Now take the L.C.M\begin{gathered} \frac{ {x}^{2} + 1 }{x} = \frac{5}{2} \\ \\ 2( {x}^{2} + 1) = 5X \\ \\ 2 {x}^{2} + 2 = 5x \\ \\ 2 {x}^{2} - 5x + 2 = 0\end{gathered}xx2+1=252(x2+1)=5x2x2+2=5x2x2−5x+2=0Here , we GET an quadratic equation in xApply middle term splitting\begin{gathered}2 {x}^{2} - 5x + 2 = 0 \\ \\ 2 {x}^{2} - 4x - x + 2 = 0 \\ \\ 2x(x - 2) - 1(x - 2) = 0 \\ \\ (2x - 1) \times (x - 2) = 0 \\ \\ \end{gathered}2x2−5x+2=02x2−4x−x+2=02x(x−2)−1(x−2)=0(2x−1)×(x−2)=0Now, EQUATE the factors to get Zero\begin{gathered}2x - 1 = 0 \\ \\ 2x = 1 \\ \\ x = \frac{1}{2} \end{gathered}2x−1=02x=1x=21Equate the other factor to zero\begin{gathered}x - 2 = 0 \\ \\ x = 2\end{gathered}x−2=0x=2So, We get \boxed{\bold{\mathsf{X = 2 \: and \: x =\frac{1}{2}}}}X=2andx=21The numbers are 2 and 1/2