Question

The sum of the digits of a two-digit number is 12.The number obtained by interchanging its digits exceeds the given number by 1ø . Find the number​

Answer 1

Given :

• Sum of digits of the two-digit no. = 12

• Increase in no. when the digits are interchanged = 10

To find :

The original number.

Solution :

Let the digits of the two-digit no. be a and b.

So the two-digit number formed in terms of a and b will be (10a + b).

And the number formed after reversing the digits will be (10b + a).

Now according to the given information , we can form two Equations and by solving them we can get the required value.

Equation (i) :-

Given the sum of digits of the two-digit number is 12 .i.e,

$\boxed{\therefore \bf{a + b = 12}}$

Hence, Equation (i) is [a + b = 12].

Equation (ii) :-

According to the Question , after the digits of the orginal number is reversed the number exceeds the original number by 12. i.e,

$:\implies \bf{(10a + b) + 10 = (10b + a)}$

$:\implies \bf{10 = (10b + a) - (10a + b)}$

$:\implies \bf{10 = 10b + a - 10a - b}$

$:\implies \bf{10 = (10b - b) + (a - 10a)}$

$:\implies \bf{10 = 9b + (-9a)}$

$:\implies \bf{10 = 9b -9a}$

$:\implies \bf{10 = 9(b - a)}$

$:\implies \bf{\dfrac{10}{9} = b - a}$

$:\implies \bf{(-a) + b = \dfrac{10}{9}}$

$\boxed{\therefore \bf{(-a) + b = \dfrac{10}{9}}}$

Hence, Equation (ii) is [(-a) + b = 4/9]

To find the orginal number :-

Now,by subtracting Equation (ii) from Equation (i) , we get :-

$:\implies \bf{(a + b) - [(-a) + b] = 12 - \dfrac{4}{9}}$

$:\implies \bf{a + b + a - b = \dfrac{108 - 4}{9}}$

$:\implies \bf{a + \not{b} + a - \not{b} = \dfrac{108 - 4}{9}}$

$:\implies \bf{a + a = \dfrac{104}{9}}$

$:\implies \bf{2a = \dfrac{104}{9}}$

$:\implies \bf{a = \dfrac{104}{9} \times \dfrac{1}{2}}$

$:\implies \bf{a = \dfrac{52}{9}}$

$\boxed{\therefore \bf{a = \dfrac{52}{9}}}$

Hence the value of a is 52/9.

Now , putting the value of a in the Equation (ii) , we get :-

$:\implies \bf{(-a) + b = \dfrac{10}{9}}$

$:\implies \bf{-\bigg(\dfrac{52}{9}\bigg) + b = \dfrac{10}{9}}$

$:\implies \bf{b = \dfrac{10}{9} + \dfrac{52}{9}}$

$:\implies \bf{b = \dfrac{10 + 52}{9}}$

$:\implies \bf{b = \dfrac{62}{9}}$

$\boxed{\therefore \bf{b = \dfrac{62}{9}}}$

Hence, the value of b is 62/9.

Now Substituting the values in the original number (in terms of a and b), we get :

$:\implies \bf{Orginal\:Number = (10a + b)}$

$:\implies \bf{Orginal\:Number = \bigg[10 \times \bigg(\dfrac{52}{9}\bigg) + \bigg(\dfrac{62}{9}\bigg)\bigg]}$

$:\implies \bf{Orginal\:Number = \bigg[\bigg(\dfrac{520}{9}\bigg) + \bigg(\dfrac{62}{9}\bigg)\bigg]}$

$:\implies \bf{Orginal\:Number = \bigg(\dfrac{520 + 62}{9}\bigg)}$

$:\implies \bf{Orginal\:Number = \bigg(\dfrac{582}{9}\bigg)}$

$\boxed{\therefore \bf{Orginal\:Number = \bigg(\dfrac{582}{9}\bigg)}}$

Hence, the original number is 582/9.