Question

Which number should be added to each of the numbers 1, 9, 13 and 45 so that the resulting numbers would be in proportion.

Answer 1

$$\begin{gathered} \mathrm{Let}x\mathrm{be}\mathrm{the}\mathrm{number}\mathrm{that}\mathrm{should}\mathrm{be}\mathrm{added}\mathrm{to}\mathrm{each}\mathrm{of}\mathrm{the}\mathrm{numbers}1,9,13\mathrm{and}45\mathrm{so}\mathrm{that}\mathrm{the}\mathrm{resulting}\mathrm{numbers}\mathrm{are}\mathrm{in}\mathrm{proportion}. \\ \mathrm{The}\mathrm{new}\mathrm{numbers}\mathrm{should}\mathrm{be}\left(1+x\right),\left(9+x\right),\left(13+x\right)\mathrm{and}\left(45+x\right)\mathrm{so}\mathrm{that}\mathrm{the}\mathrm{resulting}\mathrm{numbers}\mathrm{are}\mathrm{in}\mathrm{proportion}. \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ \frac{1+x}{9+x}=\frac{13+x}{45+x} \\ \Rightarrow \left(1+x\right)\left(45+x\right)=\left(13+x\right)\left(9+x\right) \\ \Rightarrow 45+x+45x+x^2=117+13x+9x+x^2 \\ \Rightarrow 45+46x=117+22x \\ \Rightarrow 24x=72 \\ \Rightarrow x=3 \end{gathered}$$

Hence, 3 should be added to 1, 9, 13 and 45 so that the resulting numbers are in proportion.