Let $X$ be a topological space. Define two cochain complexes $\mathcal{C}$ and $\mathcal{D}$ by
$\mathcal{C}=\{C^k(X; \mathbb{Q}), \partial^k\}, \qquad\mathcal{D}=\{C^k(X; \mathbb{R}), \partial^k\},$
where $C^k$ is the free abelian group of (singular) cochains on $X$. The inclusion of coefficients $j:\mathbb{Q}\to\mathbb{R}$ induces a chain map $i:\mathcal{C}\to\mathcal{D}$.
My question is: is the homology of the cochain complex
$\mathcal{B}:=\{C^k(X; \mathbb{R})\times C^{k-1}(X; \mathbb{Q}), \partial_j^k\},$
where $\partial_j^k(u, v)=(\partial u+j(v), -\partial v)$, equal to $H^*(X; \mathbb{R}/\mathbb{Q})$?
Of course $\mathcal{B}$ seems to be the mapping cone complex of $i:\mathcal{C}\to\mathcal{D}$, but I am not sure, since the mapping cone complex seems to be about maps between spaces.