Question

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

Answer 1

Step-by-step explanation:

$\bf\underline{\underline{\red{Given:-}}}$

• The sum of digits of a two digit number is 12.

• The number obtained by interchanging the two digits exceeds the given number by 18.

$\bf\underline{\underline{\blue{To\:Find:-}}}$

• The original number.

$\bf\underline{\underline{\green{Solution:-}}}$

Let the tens digit of the number be x

And units digit of the number of y

Then:-

The original number = 10x + y

The number obtained by interchanging the digits = 10y + x

Case 1:-

$\rm The\: sum\: of \: the \: digits\: is\: 12$

$\rm x + y = 12......(i)$

Case 2:-

The number obtained by interchanging the two digits exceeds the given number by 18.

$\rm Reversed\: number - Original \: number = 18$

$\rm \implies 10y + x - (10x + y) = 18$

$\rm \implies 10y + x - 10x - y = 18$

$\rm \implies 9y - 9x = 18$

Dividing the whole equation by 9

$\rm \implies \dfrac{9y}{9} - \dfrac{9x}{9} = \dfrac{18}{9}$

$\rm \implies y - x = 2.....(ii)$

Adding equations (i) and (ii)

$\rm \implies x + y + (y - x) = 12 + 2$

$\rm \implies x + y + y - x = 14$

$\rm \implies 2y = 14$

$\rm \implies y = \dfrac{14}{2} = 7$

Substituting y = 7 in equation (i)

$\rm \implies x + y = 12$

$\rm \implies x + 7 = 12$

$\rm \implies x = 12 - 7$

$\rm \implies x = 5$

The original number = 10x + y

= 10(5)+ (7)

= 50 + 7

= 57

$\underline{\boxed{\purple{\rm \therefore The \: original \: number = 57}}}$