Question

(i) A two digit number is 3 more than 4 times the sum of its digits. If 18 is added to the number, the digits are reversed. Find the number.

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{digit}\mathrm{at}\mathrm{units}\mathrm{place}\mathrm{be}x\mathrm{and}\text{the}\mathrm{digit}\mathrm{at}\mathrm{tens}\mathrm{place}\mathrm{be}y. \\ \mathrm{Two}\mathrm{digit}\mathrm{number}=10y+x \\ \mathrm{According}\mathrm{to}\mathrm{the}\mathrm{question}, \\ 10y+x=3+4\left(x+y\right) \\ \Rightarrow 10y+x=3+4x+4y \\ \Rightarrow 10y+x-3-4x-4y=0 \\ \Rightarrow -3x+6y=3...\left(\mathrm{i}\right) \\ \mathrm{Now},\mathrm{it}\mathrm{is}\mathrm{given}\mathrm{that}\mathrm{if}18\mathrm{is}\mathrm{added}\mathrm{to}\mathrm{the}\mathrm{number},\text{the}\mathrm{digits}\mathrm{will}\text{be}\mathrm{reversed}. \\ 18+10y+x=10x+y \\ \Rightarrow 9y-9x=-18 \\ \Rightarrow -x+y=-2 \\ \Rightarrow x-y=2...\left(\mathrm{ii}\right) \\ \mathrm{Multiplying}\mathrm{by}3\mathrm{in}\mathrm{eq}\left(\mathrm{ii}\right),\text{we get:} \\ 3x-3y=6...\left(\mathrm{iii}\right) \\ \mathrm{Adding}\left(\mathrm{i}\right)\&\left(\mathrm{iii}\right),\mathrm{we}\mathrm{get}: \\ -3x+6y=3 \\ \underline{3x-3y=6} \\ 3y=9 \\ \therefore y=3 \\ S\mathrm{ubstituting}y=3\mathrm{in}\mathrm{eq}\left(\mathrm{i}\right),\mathrm{we}\mathrm{get}: \\ -3x+6\times 3=3 \\ -3x=-15 \\ \therefore x=5 \\ \mathrm{Hence},\mathrm{the}\mathrm{number}\mathrm{would}\mathrm{be}35. \end{gathered}$$