3.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?

Let unit place digit be x.
Ten’s place digit y= 9 – x[Sum of both digits: x+y=9]

Original number = x + 10(9 – x)

Condition I: 10x + (9 – x) (Interchanging the digits)

Condition II: New number = original number + 27

\(\Rightarrow\) 10x + (9 – x) = x + 10(9 – x) + 27

\(\Rightarrow\) 10x + 9 – x = x + 90 – 10x + 27 (solving the brackets)

\(\Rightarrow\) 9x + 9 = -9x + 117 (Transposing 9x to LHS and 9 to RHS)

\(\Rightarrow\) 9x + 9x = 117 – 9

\(\Rightarrow\) 18x = 108

\(\Rightarrow\) x = 108 ÷ 18 (Transposing 18 to RHS)

\(\Rightarrow\) x = 6

Hence we have,

Unit place digit = 6

Tens place digit = 9 – 6 = 3

Thus, the required number = 6 + 3 × 10 = 6 + 30 = 36