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Minimal example, multiplying by 0.5 makes the command fail :
\documentclass{beamer}
\usepackage{graphicx}
\begin{document}
\newcommand{\dontwork}[1]{\includegraphics[width=0.5*(#1)\textwidth]{frog.jpg}}
\newcommand{\work}[1]{\includegraphics[width=#1\textwidth]{frog.jpg}}
\begin{frame}{Frame 1}
\dontwork{1}
\end{frame}
\end{document}
I get this error :
Use \dimexpr (provided by the e-TeX extensions whose availability is standard in LaTeX nowadays):
\includegraphics[width=\dimexpr.5\textwidth*(#1)\relax]{frog.jpg}
Be aware that in case of multiplication the formal syntax of \dimexpr requires the dimension to come first and the scaling-factor to come last, nested in parentheses in case it is formed by a numerical expression. E.g.,
\dimexpr(.5\textwidth)*(#1)\relax
does work if #1 forms a valid numerical expression while
\dimexpr(#1)*(.5\textwidth)\relax
does not.
If #1 shall not necessarily be an integer, then scale the \dimexpr itself:
\includegraphics[width=#1\dimexpr.5\textwidth\relax]{frog.jpg}
or
\includegraphics[width=.5\dimexpr#1\textwidth\relax]{frog.jpg}
You could use
\includegraphics[width=#1\dimexpr0.5\textwidth\relax]{...}
Alhough specifying a fraction of half \textwidth seems a very weird interface, are you wanting a fraction of \columnwidth ??
\newcommand{\adaptgraphics}[2][1]{\includegraphics[width=#1\textwidth,height=#1\dimexpr0.77\textheight,keepaspectratio]{figures/#2}}
I got an answer on a latex discord :
\newcommand{\flatfrog}[1]{\includegraphics[width=\dimexpr 0.33\textwidth*(#1)\relax]{frog.jpg}}
Works, \dimexpr makes the evaluation preserving the dimension of \textwidth which is needed