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 digit be x

then tens digit is 9-x

the number is : 10(9-x) + x
= 90 - 9x

the number reversed is 10x + 9-x
= 9x + 9

9x+9 - (90-9x) = 27
18x - 81 = 27
18x = 108
x = 6

unit digit is 6
tens digit is 3
The number is 36

Dl vv

To find the two-digit number, let's assume the tens digit as x and the units digit as y.

According to the given condition, the sum of the digits is 9, so we can write the equation:

x + y = 9 -- Equation 1

When we interchange the digits, the resulting new number is greater than the original number by 27. This means that the new number is obtained by switching the tens and units digits.

So, the new number can be written as 10y + x (units digit becomes tens digit, and tens digit becomes units digit), and the original number is 10x + y.

According to the condition, the new number is greater than the original number by 27, so we can write the equation:

10y + x = 10x + y + 27 -- Equation 2

Now, we have a system of two equations (Equation 1 and Equation 2) with two unknowns (x and y). We can solve this system of equations to find the values of x and y, which will give us the original two-digit number.

Let's solve the system of equations:

From Equation 1, we can rewrite it as:
x = 9 - y

Substituting this in Equation 2:
10y + (9 - y) = 10(9 - y) + y + 27

Simplifying this equation:
10y + 9 - y = 90 - 10y + y + 27
9y + 9 = 90 + 27
9y + 9 = 117
9y = 108
y = 12

Substituting y = 12 in Equation 1:
x + 12 = 9
x = 9 - 12
x = -3

However, we are looking for a two-digit number, so we discard the solution where x is negative. Therefore, the two-digit number is with x = 9 and y = 12.

Therefore, the two-digit number is 9-12 which is 21.