Question

A number consists of two digits whose sum is 9. If 27 is added to the number, the digits change their places. Find the number.

Answer 1

Let the digit at the tens place be x and digit at the units place be y.
The number is 10x + y.

According to the question,
$$\begin{gathered} x+y=9...\left(1\right) \\ 10x+y+27=10y+x...\left(2\right) \\ 10x+y+27=10y+x \\ \Rightarrow 10x+y-10y-x=-27 \\ \Rightarrow 9x-9y=-27 \\ \Rightarrow 9\left(x-y\right)=-27 \\ \Rightarrow x-y=-3 \\ \Rightarrow x=-3+y...\left(3\right) \\ \mathrm{Substituting}\mathrm{the}\mathrm{value}\mathrm{of}x\mathrm{in}\left(1\right),\mathrm{we}\mathrm{get} \\ -3+y+y=9 \\ \Rightarrow 2y=9+3 \\ \Rightarrow 2y=12 \\ \Rightarrow y=6...\left(4\right) \\ \mathrm{Substituting}\mathrm{the}\mathrm{value}\mathrm{of}y\mathrm{in}\left(3\right),\mathrm{we}\mathrm{get} \\ x=-3+6 \\ \Rightarrow x=3 \\ \mathrm{Hence},\mathrm{the}\mathrm{number}\mathrm{is}36. \end{gathered}$$