A number consisting of two digits is seven times the sum of its digits

1.	A number consisting of two digits is seven times the sum of its digits. When 27 is subtracted from the number, the digits are reversed. Find the number.
Answer» Let the tens and the units digits of the required number be x and y, respectively. Required number = (10x + y) 10x + y = 7(x + y) 10x + 7y = 7x + 7y or 3x – 6y = 0 ……….(i) Number obtained on reversing its digits = (10y + x) (10x + y) - 27 = (10y + x) ⇒10x – x + y – 10y = 27 ⇒9x – 9y = 27 ⇒9(x – y) = 27 ⇒x – y = 3 ……..(ii) On multiplying (ii) by 6, we get: 6x – 6y = 18 ………(iii) On subtracting (i) from (ii), we get: 3x = 18 ⇒ x = 6 On substituting x = 6 in (i) we get 3 × 6 – 6y = 0 ⇒ 18 – 6y = 0 ⇒ 6y = 18 ⇒ y = 3 Number = (10x + y) = 10 × 6 + 3 = 60 + 3 = 63 Hence, the required number is 63.

A number consisting of two digits is seven times the sum of its digits. When 27 is subtracted from the number, the digits are reversed. Find the number.

Answer»

Let the tens and the units digits of the required number be x and y, respectively.

Required number = (10x + y)

10x + y = 7(x + y)

10x + 7y = 7x + 7y or 3x – 6y = 0 ……….(i)

Number obtained on reversing its digits = (10y + x)

(10x + y) - 27 = (10y + x)

⇒10x – x + y – 10y = 27

⇒9x – 9y = 27

⇒9(x – y) = 27

⇒x – y = 3 ……..(ii)

On multiplying (ii) by 6, we get:

6x – 6y = 18 ………(iii)

On subtracting (i) from (ii), we get:

3x = 18

⇒ x = 6

On substituting x = 6 in (i) we get

3 × 6 – 6y = 0

⇒ 18 – 6y = 0

⇒ 6y = 18

⇒ y = 3

Number = (10x + y) = 10 × 6 + 3 = 60 + 3 = 63

Hence, the required number is 63.