The sum of the two-digit number is 9. When we interchange the digits the new number is 27 greater than the earlier number. Find the number.
A two-digit number 'ab' can be represented as a sum of its respective place values i.e., ab = 10a + b
Answer: The number whose sum of the digits is 9, and when we interchange the digits the new number is 27 greater than the earlier number, is 36.
Let's find the number by applying the given conditions in the question as follows
Let the digits of the original number be x and y
Hence, the original number is 10x + y (Assuming x to be the ten's digit and y to be the one's digit)
After interchanging the digits the new number will be 10y + x (After reversing, y becomes the ten's digit and x becomes the one's digit)
Condition 1: Sum of the digits is 9 ⇒ x + y = 9 --------- equation (i)
Condition 2: The number obtained by interchanging the digits is 27 greater than the earlier number.
⇒ New number = 27 + original number
⇒ 10y + x = 27 + (10x + y)
⇒ 10y + x = 27 + 10x + y
⇒ 10y - y + x - 10x = 27
⇒ 9y - 9x = 27
⇒ y - x = 3 -------- equation (ii)
By adding equation(i) and equation(ii):
x + y + y - x = 9 + 3
⇒ 2y = 12
⇒ y = 6
From equation(i): x + 6 = 9
⇒ x = 9 - 6 = 3
⇒ x = 3 and y = 6
⇒ The required number is 10x + y = 10 × 3 + 6 = 30 + 6 = 36
Thus, the required two-digit number is 36.