Explain the average rate of reaction for a hypothetical $R \longrightarrow P$ reaction.

Consider a hypothetical reaction where the volume of the system remains constant.
Reaction: $R \longrightarrow P$
Suppose at time $t_{1}$,the concentration of reactant $R$ is $[R]_{1}$.
Suppose at time $t_{2}$,the concentration of reactant $R$ is $[R]_{2}$.
The change in time is $\Delta t = t_{2} - t_{1}$.
The change in concentration of the reactant is $\Delta[R] = [R]_{2} - [R]_{1}$.
The rate of decrease in the concentration of reactant $R$ is given by:
Average Rate $= -\frac{\Delta[R]}{\Delta t} = -\frac{[R]_{2} - [R]_{1}}{t_{2} - t_{1}} \dots (i)$
Similarly,if at time $t_{1}$ the concentration of product $P$ is $[P]_{1}$ and at time $t_{2}$ it is $[P]_{2}$,the increase in concentration of the product is $\Delta[P] = [P]_{2} - [P]_{1}$.
The rate of formation of product $P$ is given by:
Average Rate $= +\frac{\Delta[P]}{\Delta t} = +\frac{[P]_{2} - [P]_{1}}{t_{2} - t_{1}} \dots (ii)$
Equating $(i)$ and $(ii)$,the average rate of reaction is:
Average Rate $= -\frac{\Delta[R]}{\Delta t} = +\frac{\Delta[P]}{\Delta t}$