Question

(viii) A two-digit number is 3 more than six times the sum of its digits. If 18 is added to the number obtained by interchanging the digits, we get the original number. Find the number.

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{digit}\mathrm{at}\mathrm{units}\mathrm{place}=x \\ \mathrm{Digit}\mathrm{at}\mathrm{tens}\mathrm{place}=y \\ \text{So, the n}\mathrm{umber}\mathrm{will}\mathrm{be}=10y+x \\ \mathrm{According}\mathrm{to}\mathrm{the}\mathrm{question}, \\ 10y+x=3+6\left(x+y\right) \\ \Rightarrow 10y+x=3+6x+6y \\ \Rightarrow -5x+4y=3...\left(1\right) \\ \mathrm{Number}\mathrm{obtained}\mathrm{by}\mathrm{interchanging}\mathrm{the}\mathrm{digits}=10y+x \\ \mathrm{According}\mathrm{to}\mathrm{the}\mathrm{question}, \\ 10x+y+18=10y+x \\ \Rightarrow 9x-9y=-18 \\ \Rightarrow x-y=-2...\left(2\right) \\ \mathrm{Multiplying}\mathrm{eq}\left(2\right)\mathrm{by}4,\mathrm{we}\mathrm{get}: \\ 4x-4y=-8...\left(3\right) \\ \mathrm{Adding}\mathrm{eq}\left(1\right)\mathrm{and}\mathrm{eq}\left(3\right),\text{we get:} \\ -5x+4y=3 \\ \underline{4x-4y=-8} \\ -x=-5 \\ \therefore x=5 \\ \mathrm{Substituting}x=5\mathrm{in}\mathrm{eq}\left(2\right), \\ 5-y=-2 \\ \Rightarrow y=7 \\ \mathrm{Hence},\mathrm{the}\mathrm{number}=10y+x=10\times 7+5=75. \end{gathered}$$