We find the greatest value $\lambda$ and the least value $\mu$ such that the double inequality \begin{align*} C(ambda a+(1-ambda)b,ambda b+(1-ambda)a)&<lpha A(a,b)+(1-lpha)T(a,b) &<C(\mu a+(1-\mu)b,\mu b+(1-\mu)a) \end{align*} holds for all $\alpha\in(0,1)$ and $a,b>0$ with $a\neq b$, where $C(a,b)$, $A(a,b)$, and $T(a,b)$ denote respectively the contraharmonic, arithmetic, and Toader means of two positive numbers $a$ and $b$.