question_answer 21)

                Sum of the digits of a two-digit number is 9. When we interchange the digits, it is found that the resulting new number is greater than the original number by 27. What is the two-digit number?

Answer:

                Let the units digit of the two-digit number be \[x\]. Then, the tens digit of the two-digit number \[=9-x\] |\[\because \] Sum of the digits of the two-digit number is 9 \[\therefore \] Original number               \[=10(9-x)+x\] \[=90\,-10x+x\] \[=90-9x\] When we interchange the digits, then Units digit                            \[=9-x\] and,   tens digit                 \[=x\] \[\therefore \] Resulting number            \[=10x+(9-x)\] \[=9x+9\] According to the question, \[(9x+9)=(90-9x)+27\] \[\Rightarrow \]               \[9x+9x=90+27-9\]                          | Transposing - 9x to LHS and 9 to RHS \[\Rightarrow \]               \[18x=108\] \[\Rightarrow \]               \[x=\frac{108}{18}=6\]                   | Dividing both sides by 18           \[\Rightarrow \]               \[9-x=9-6=3\] Hence, the required two-digit number is 36. Check: \[3+6=9\] \[63=36+27\] Hence, the result is verified.