The sum of a two digit number and the number formed by interchanging the digits is 110. If 10 is subtracted from the first number, the new number is 4 more than 5 times the sums of the digits of the first number. Find the first number.

#### Solution

Let the ten’s digit be x and the unit digit be y

The number is 10x + y

If the digits are interchanged

The new number is 10y + x

By the given first condition

10x + y + 10y + x = 110

11x + 11y = 110

x + y = 10 → (1) ...(Divided by 11)

Again by the given second condition

10x + y – 10 = 5(x + y) + 4

10x + y – 10 = 5x + 5y + 4

5x – 4y = 14 → (2)

(1) × 5 ⇒ 5x + 5y = 50 → (3)

(2) × 1 ⇒ 5x – 4y = 14 → (2)

(3) – (2) ⇒ 9y = 36

y = `36/9`

= 4

Substitute the value of y = 4 in (1)

x + y = 10

x + 4 = 10

x = 10 – 4

= 6

∴ The number is (10 × 6 + 4) = 64