The sum of the digits of a two-digit number is 12. The number obtained

1.	The sum of the digits of a two-digit number is 12. The number obtained by interchanging its digits exceeds the given number by 18. Find the number.
Answer» Let the two digit number is yx = x + 10y. (∵ Any two digit number ab is written as ab = 10a + b) Given that the sum of digits of the number is 12. ∴ x + y = 12 … (1) The number obtained by interchanging its digits is xy = y + 10x. Given that, the number obtained by interchanging its digits exceeds the given number by 18. i.e. y + 10x = x + 10y +18 ⇒ 9x – 9y = 18 … (2) Now multiplying equation (1) by 9, we get 9x + 9y = 108 … (3) Now, adding equation (2) and (3), we get (9x – 9y) + (9x + 9y) = 18 + 108. ⇒18x = 126 ⇒ x = \(\frac{126}{18}\) = 7. Now putting x = 7 in equation (1), we get 7 + y =12 ⇒ y = 12 – 7 = 5. Hence, the number is x + 10y = 7 + 10 × 5 = 7 + 50 = 57. Hence, the number is 57.

The sum of the digits of a two-digit number is 12. The number obtained by interchanging its digits exceeds the given number by 18. Find the number.

Answer»

Let the two digit number is yx = x + 10y.

(∵ Any two digit number ab is written as ab = 10a + b)

Given that the sum of digits of the number is 12.

∴ x + y = 12 … (1)

The number obtained by interchanging its digits is xy = y + 10x.

Given that, the number obtained by interchanging its digits exceeds the given number by 18.

i.e. y + 10x = x + 10y +18

⇒ 9x – 9y = 18 … (2)

Now multiplying equation (1) by 9, we get

9x + 9y = 108 … (3)

Now, adding equation (2) and (3), we get

(9x – 9y) + (9x + 9y) = 18 + 108.

⇒18x = 126

⇒ x = \(\frac{126}{18}\) = 7.

Now putting x = 7 in equation (1), we get 7 + y =12 ⇒ y = 12 – 7 = 5.

Hence, the number is x + 10y = 7 + 10 × 5 = 7 + 50 = 57.

Hence, the number is 57.

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