Question

In a recent​ year, companies spent a total of ​$88.9 billion on​ newspaper, television, and radio ads. The total amount spent on television and radio ads was only ​$3.1 billion more than the amount spent on newspaper ads alone. The amount spent on newspaper ads was​ $5.4 billion more than what was spent on television ads. How much was spent on each form of​ advertising? ​(​Hint: Let the variables represent numbers of billions of​ dollars.) How much was spent on newspaper​ ads? ​$ nothing billion How much was spent on television​ ads? ​$ nothing billion How much was spent on radio​ ads? ​$ nothing billion

Answer 1

$\bf \begin{cases} t=\textit{tv ads}\\ r=\textit{radio ads}\\ n=\textit{newspaper ads} \end{cases}~\hspace{7em}t+r+n=88.9 \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{tv, radio ads 3.1 more than newspaper}}{t+r=n+3.1}~\hfill \stackrel{\textit{newspaper ads 5.4 more than tv's}}{\boxed{n} = t+5.4}$

$\bf \stackrel{\textit{substituting on the 2nd equation}}{t+r=\boxed{t+5.4}+3.1}\implies ~~\begin{matrix} t \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~+r=~~\begin{matrix} t \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~+8.5\implies \blacktriangleright r=8.5 \blacktriangleleft \\\\\\ \stackrel{\textit{we know that}}{t+r+n=88.9}\implies t+8.5+n=88.9\implies t+n=80.4$

$\bf \boxed{n}=80.4-t~\hspace{7em}\stackrel{\textit{substituting on the 3rd equation}}{n=t+5.4\implies \boxed{80.4-t}=t+5.4} \\\\\\ 75-t=t\implies 75=2t\implies \cfrac{75}{2}=t\implies \blacktriangleright 37.5=t \blacktriangleleft \\\\\\ \stackrel{\textit{since we know that}}{n = t+5.4}\implies n=37.5+5.4\implies \blacktriangleright n=42.9 \blacktriangleleft$