sum of digits of two digit number is 9.When the digits are interchange

1.	sum of digits of two digit number is 9.When the digits are interchanged the new number is greater than the original number by 9.Find the original number.
Answer» Given : Sum of digits of TWO digit number is 9. When the digits are interchanged the new number is greater than the original number by 9. To FIND : The original number. SOLUTION : Let the digit at the tens place be x. Let the digit at the units place be y. Original Number = (10x + y) Case 1 : The sum of tens digit and units digit is 9. Equation : $\implies$ $\sf{x+y=9}$ $\sf{x=9-y\:\:\:\:\bold{1}}$ Case 2 : When the digits of the two digit number are interchanged, the new number is greater than original number by 9. Reversed Number = (10y +x) Equation : $\implies$ $\sf{10y+x=10x+y+9}$ $\implies$ $\sf{10y-y=10x-x+9}$ $\implies$ $\sf{9y=9x+9}$ $\implies$ $\sf{9x+9=9y}$ $\implies$ $\sf{9x-9y=-9}$ $\implies$ $\sf{9(9-y) -9y=-9}$ $\implies$ $\sf{81-9y-9y=-9}$ $\implies$ $\sf{-18y=-9-81}$ $\implies$ $\sf{-18y=-90}$ $\implies$ $\sf{y=\dfrac{-90}{-18}}$ $\implies$ $\sf{y=\dfrac{90}{18}}$ $\implies$ $\sf{y=<klux>5</klux>}$ Substitute, y = 5 in equation (1), $\implies$ $\sf{x=9-y}$ $\implies$ $\sf{x=9-5}$ $\implies$ $\sf{x=4}$ $\large{\boxed{\bold{Ten's\:digit\:=\:x\:=\:4}}}$ $\large{\boxed{\bold{Unit's\:digit\:=\:y\:=\:5}}}$ $\large{\boxed{\bold{\purple{Original\:Number\:=\:10x+y=10(4)+5=40+5}}}}$