Question

The sum of the digits of a 2-digit number is 11. The number obtained by
interchanging the digits exceeds the original number by 27. Find the number.​

Answer 1

Answer:

$\sf{The \ number \ is \ 47.}$

Given:

$\sf{\leadsto{The \ sum \ of \ digits \ of \ a \ two \ digit}}$

$\sf{number \ is \ 11.}$

$\sf{\leadsto{The \ number \ obtained \ by \ interchanging}}$

$\sf{the \ digits \ exceed \ the \ original \ number}$

$\sf{by \ 27.}$

To find:

$\sf{The \ number.}$

Solution:

$\sf{Let \ ten's \ place \ of \ a \ two \ digit \ number}$

$\sf{be \ x \ and \ unit's \ place \ be \ y.}$

$\sf{According \ to \ the \ first \ condition.}$

$\sf{x+y=11...(1)}$

$\sf{Original \ number=10x+y}$

$\sf{Number \ with \ reversed \ digits=10y+x}$

$\sf{According \ to \ the \ second \ condition.}$

$\sf{10y+x=10x+y+27}$

$\sf{\therefore{-9x+9y=27}}$

$\sf{\therefore{9(-x+y)=27}}$

$\sf{\therefore{-x+y=3...(2)}}$

$\sf{Adding \ equations \ (1) \ and \ (2), \ we \ get}$

$\sf{x+y=11}$

$\sf{+}$

$\sf{-x+y=3}$

_______________

$\sf{2y=14}$

$\sf{\therefore{y=\dfrac{14}{2}}}$

$\boxed{\sf{\therefore{y=7}}}$

$\sf{Substitute \ y=7 \ in \ equation (1), \ we \ get}$

$\sf{x+7=11}$

$\boxed{\sf{\therefore{x=4}}}$

$\sf{Original \ number=10x+y=10(4)+7=47}$

$\sf\purple{\tt{\therefore{The \ number \ is \ 47.}}}$