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 the number at ones place of two digit number = a
According to question, the sum of digits of given two digit number = 9
i.e. Digit at tens place + digit at ones place = 9
Or, Digit at tens place + a = 9

By transposing ‘a’ to RHS, we get
Digit at tens place = 9 – a
Thus, the number = 10(9 – a) + a
After interchange of digit, the number = 10a + (9 – a)

Since, number obtained after interchange of digit is greater than the original number by 27
Therefore, New number – 27 = Original number

Here, we have original number = 10(9 – a) + a
And, new number = 10a + (9 – a)
⇒ 10a + (9 – a) – 27 = 10 (9 – a) + a
⇒ 10a + 9 – a – 27 = 90 – 10a + a
⇒ 10a – a + 9 – 27 = 90 – 9a
⇒ 9a – 18 = 90 – 9a

By transposing 18 to RHS, we get
9a = 90 – 9a + 18
By transposing – 9a to LHS, we get
9a + 9a = 90 + 18
⇒18 a = 108

After dividing both sides by 18, we get

Since, digit at tens place = 9 –a
Thus, by substituting the value of a, we get
The number at tens place = 9 – a = 9 – 6 = 3
Thus, number at tens place = 3
And number at ones place = a = 6

Thus, the number = 36