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
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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
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[tex]\displaystyle\large\underline{\sf\red{Given}}[/tex]
✭ 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
[tex]\displaystyle\large\underline{\sf\blue{To \ Find}}[/tex]
◈ The original number?
[tex]\displaystyle\large\underline{\sf\gray{Solution}}[/tex]
So here let the original number be
Then on interchanging the digits the number becomes,
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[tex]\underline{\bigstar\:\textsf{According to the given Question :}}[/tex]
Sum of digits of the original number is 11,so,
➝ [tex]\displaystyle\sf x+y = 11\:\:\:-eq(1)[/tex]
Also the new Number is 27 more than the original number,
➳ [tex]\displaystyle\sf 10x+y+27 = 10y+x[/tex]
➳ [tex]\displaystyle\sf 10x-x+y-10y = -27[/tex]
➳ [tex]\displaystyle\sf 9x-9y = 27[/tex]
➳ [tex]\displaystyle\sf 9(x-y) = -27[/tex]
➳ [tex]\displaystyle\sf x-y = \dfrac{-27}{9}[/tex]
➳ [tex]\displaystyle\sf -x+y = 3\:\:\: -eq(2)[/tex]
On subtracting eq(2) from eq(1)
»» [tex]\displaystyle\sf (x+y)-(-x+y) = 11-3[/tex]
»» [tex]\displaystyle\sf x+y+x-y = 8[/tex]
»» [tex]\displaystyle\sf 2x = 8[/tex]
»» [tex]\displaystyle\sf x = \dfrac{8}{2}[/tex]
»» [tex]\displaystyle\sf \orange{x = 4}[/tex]
Substituting the value of x in eq(1)
›› [tex]\displaystyle\sf x+y = 11[/tex]
›› [tex]\displaystyle\sf 4+y = 11[/tex]
›› [tex]\displaystyle\sf y = 11-4[/tex]
›› [tex]\displaystyle\sf \pink{y = 7}[/tex]
[tex]\displaystyle\therefore\:\underline{\sf The \ Number \ will \ be \ 47}[/tex]
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Answer:
[tex]\sf{The \ number \ is \ 47.}[/tex]
Given:
[tex]\sf{\leadsto{The \ sum \ of \ digits \ of \ a \ two \ digit}}[/tex]
[tex]\sf{number \ is \ 11.}[/tex]
[tex]\sf{\leadsto{The \ number \ obtained \ by \ interchanging}}[/tex]
[tex]\sf{the \ digits \ exceed \ the \ original \ number}[/tex]
[tex]\sf{by \ 27.}[/tex]
To find:
[tex]\sf{The \ number.}[/tex]
Solution:
[tex]\sf{Let \ ten's \ place \ of \ a \ two \ digit \ number}[/tex]
[tex]
\sf{be \ x \ and \ unit's \ place \ be \ y.}
[/tex]
[tex]\sf{According \ to \ the \ first \ condition.}[/tex]
[tex]\sf{x+y=11}...(1)[/tex]
[tex]\sf{Original \ number=10x+y}[/tex]
[tex]\sf{Number \ with \ reversed \ digits=10y+x} \\
\sf{According \ to \ the \ second \ condition.} \\
\sf{10y+x=10x+y+27}[/tex]
[tex]\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}
[/tex]
[tex]
\sf{-x+y=3}
[/tex]
_______________
[tex]\sf{2y=14} \\ \\
\sf{\therefore{y=\dfrac{14}{2}}}
[/tex]
[tex]\boxed{\sf{\therefore{y=7}}} [/tex]
[tex]\sf{Substitute \ y=7 \ in \ equation (1), \ we \ get} \\
\sf{x+7=11} \\
\boxed{\sf{\therefore{x=4}}} [/tex]
∴
[tex]\sf{Original \ number=10x+y=10(4)+7=47}[/tex]
[tex]\sf\pink{\tt{\therefore{The \ number \ is \ 47.}}}[/tex]