A number is made up of two digits. the sum of digits is 11, if the digits are interchanged, the original number is increased by 9. find the number

56

Step-by-step explanation:

let the 2 digits be a and b

Create 2 equations, solve simultaneously for a and b

a + b = 11 → (1) (sum of digits)

The original number is ab, that is 10a + b (considering place value)

Interchanging the 2 digits gives ba, that is 10b + a, thus

10b + a = 10a + b + 9 (original number increased by 9)

Subtract b from both sides

9b + a = 10a + 9 (subtract 10a from both sides)

9b - 9a = 9 → (2)

Rearranging (1) expressing a in terms of b

a = 11 - b

Substitute a = 11 - b into (2)

9b - 9 (11 - b) = 9 ← distribute and simplify left side

9b - 99 + 9b = 9

18b - 99 = 9 (add 99 to both sides)

18b = 108 (divide both sides by 18)

b = 6

Substitute b = 6 into (1)

a + 6 = 11 ⇒ a = 5

The original number is 56