The sum of two numbers is 75 and their difference is 25. The product of the two numbers is :

Answer: Option B

Let the numbers be a and b
According to the question,
$$\begin{array}{rl}& a+b=75\\ & a-b=25\\ & \because {\left(a+b\right)}^2-{\left(a-b\right)}^2=4ab\\ & \Rightarrow 75^2-25^2=4ab\\ & \Rightarrow 4ab=\left(75+25\right)\left(75-25\right)\\ & \left[\because a^2-b^2=\left(a+b\right)\left(a-b\right)\right]\\ & \Rightarrow 4ab=100\times 50\\ & \Rightarrow ab=\frac{100\times 50}{4}\\ & \Rightarrow ab=1250\end{array}$$