$\begin{align}  & \text{Let the total allowance be }x \\ & \tfrac{1}{5}\text{ of }x\text{ was spent on books,} \\ & \text{The amount left will be }\tfrac{4}{5}x \\ & \text{He spent }\tfrac{1}{3}\text{ of }\tfrac{4}{5}x\text{ on food which is }\tfrac{1}{3}\times \tfrac{4x}{5}=\tfrac{4x}{15} \\ & \text{The amount left left will be }\tfrac{4}{5}x-\tfrac{4}{15}x=\tfrac{8x}{15} \\ & \text{The fraction kept is }\tfrac{8}{15} \\\end{align}$