Question

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

Answer 1

$\displaystyle\large\underline{\sf\red{Given}}$

✭ Sum of the digits of a two digit number is 11

✭ The Number obtained on interchanging the digits is 27 more than the original number

$\displaystyle\large\underline{\sf\blue{To \ Find}}$

◈ The original number?

$\displaystyle\large\underline{\sf\gray{Solution}}$

So here let the original number be

• 10x+y

Then on interchanging the digits the number becomes,

• 10y+x

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$\underline{\bigstar\:\textsf{According to the given Question :}}$

Sum of digits of the original number is 11,so,

➝ $\displaystyle\sf x+y = 11\:\:\:-eq(1)$

Also the new Number is 27 more than the original number,

➳ $\displaystyle\sf 10x+y+27 = 10y+x$

➳ $\displaystyle\sf 10x-x+y-10y = -27$

➳ $\displaystyle\sf 9x-9y = 27$

➳ $\displaystyle\sf 9(x-y) = -27$

➳ $\displaystyle\sf x-y = \dfrac{-27}{9}$

➳ $\displaystyle\sf -x+y = 3\:\:\: -eq(2)$

On subtracting eq(2) from eq(1)

»» $\displaystyle\sf (x+y)-(-x+y) = 11-3$

»» $\displaystyle\sf x+y+x-y = 8$

»» $\displaystyle\sf 2x = 8$

»» $\displaystyle\sf x = \dfrac{8}{2}$

»» $\displaystyle\sf \orange{x = 4}$

Substituting the value of x in eq(1)

›› $\displaystyle\sf x+y = 11$

›› $\displaystyle\sf 4+y = 11$

›› $\displaystyle\sf y = 11-4$

›› $\displaystyle\sf \pink{y = 7}$

$\displaystyle\therefore\:\underline{\sf The \ Number \ will \ be \ 47}$

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Answer 2

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:

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

$</p><p>\sf{be \ x \ and \ unit's \ place \ be \ y.}</p><p>$

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

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

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

$\sf{Number \ with \ reversed \ digits=10y+x} \\ </p><p>\sf{According \ to \ the \ second \ condition.} \\ </p><p>\sf{10y+x=10x+y+27}$

$\sf{\therefore{-9x+9y=27}} \\ </p><p></p><p>\sf{\therefore{9(-x+y)=27}} \\ </p><p></p><p>\sf{\therefore{-x+y=3...(2)}} \\ </p><p></p><p>\sf{Adding \ equations \ (1) \ and \ (2), \ we \ get}</p><p>\sf{x+y=11}</p><p>$

$</p><p></p><p>\sf{-x+y=3}</p><p>$

_______________

$\sf{2y=14} \\ \\ </p><p></p><p>\sf{\therefore{y=\dfrac{14}{2}}}</p><p>$

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

$\sf{Substitute \ y=7 \ in \ equation (1), \ we \ get} \\ </p><p>\sf{x+7=11} \\ </p><p></p><p>\boxed{\sf{\therefore{x=4}}}$

∴

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

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