Question

Auntie Ella is three times as old as her niece today. 4 years ago she was 4 times as old as her niece. Find the sum of their present ages.

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{present}\mathrm{age}\mathrm{of}\mathrm{niece}=\mathrm{x}\mathrm{yrs} \\ \mathrm{Then}\mathrm{present}\mathrm{age}\mathrm{of}\mathrm{Aunty}=3\mathrm{x}\mathrm{yrs} \\ \mathbf{4}\mathbf{}\mathbf{years}\mathbf{}\mathbf{ago}\mathbf{}\mathbf{:} \\ \mathrm{age}\mathrm{of}\mathrm{niece}=\left(\mathrm{x}-4\right)\mathrm{yrs} \\ \mathrm{age}\mathrm{of}\mathrm{Aunty}=\left(3\mathrm{x}-4\right)\mathrm{yrs} \\ \mathrm{Now},\mathrm{according}\mathrm{to}\mathrm{question}: \\ 3\mathrm{x}-4=4\left(\mathrm{x}-4\right) \\ \Rightarrow 3\mathrm{x}-4=4\mathrm{x}-16 \\ \Rightarrow 3\mathrm{x}-4\mathrm{x}=-16+4 \\ \Rightarrow -\mathrm{x}=-12 \\ \Rightarrow \mathrm{x}=12 \\ \mathrm{So},\mathrm{present}\mathrm{age}\mathrm{of}\mathrm{niece}=12\mathrm{yrs} \\ \mathrm{present}\mathrm{age}\mathrm{of}\mathrm{Aunty}=36\mathrm{yrs} \\ \mathrm{sum}\mathrm{of}\mathrm{present}\mathrm{ages}=12+36=48\mathrm{yrs} \end{gathered}$$