Question

.The sum of the digits of a two digit number is 7. The number obtained by interchanging the digits exceeds the original number by 27. Find the original number.

Answer 1

Solution :

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

The sum of the digits of a two digit number is 7. The number obtained by interchanging the digits exceeds the original number by 27.

$\bf{\red{\underline{\bf{To\:find\::}}}}$

The original number.

$\bf{\red{\underline{\bf{Explanation\::}}}}$

Let the ten's place digit be r

Let the one's place digit be m

$\boxed{\bf{The\:original\:number=10r+m}}}}}\\\boxed{\bf{The\:reversed\:number=10m+r}}}}}$

A/q

$\longrightarrow\sf{r+m=7}\\\\\longrightarrow\sf{m=7-r.................(1)}$

&

$\longrightarrow\sf{10m+r=10r+m+27}\\\\\longrightarrow\sf{10m-m+r-10r=27}\\\\\longrightarrow\sf{9m-9r=27}\\\\\longrightarrow\sf{9(m-r)=27}\\\\\longrightarrow\sf{m-r=\cancel{\dfrac{27}{9} }}\\\\\longrightarrow\sf{m-r=3}\\\\\longrightarrow\sf{7-r-r=3\:\:\:[from(1)]}\\\\\longrightarrow\sf{7-2r=3}\\\\\longrightarrow\sf{-2r=3-7}\\\\\longrightarrow\sf{-2r=-4}\\\\\longrightarrow\sf{r=\cancel{\dfrac{-4}{-2} }}\\\\\longrightarrow\sf{\purple{r=2}}$

Putting the value of r in equation (1),we get;

$\longrightarrow\sf{m=7-2}\\\\\longrightarrow\sf{\purple{m=5}}$

Thus;

$\underbrace{\sf{The\:original\:number\:=(10r+m)=10(2)+5=20+5=\boxed{25}}}}}$

Answer 2

$\bigstar$ Solution :

$\bigstar$ Given Data :

$\star$ The sum of the digits of a two digit number is 7. The number obtained by interchanging the digits exceeds the original number by 27.

$\bigstar$ To Find :

$\star$ The original two digit number.

$\bigstar$ Explanation :

$\star$ Let the ten's place digit be x

$\star$ Let the one's place digit be y

$\star$ So the original number be ' 10x + y '

$\star$ and Reversed number be ' 10y + x '

$\star$ A/c to the 1st condition ,

=> x + y = 7 ...(1)

$\star$ A/c to the 2nd condition,

=> 10y + x = 10x + y + 27

=> 9y = 9x + 27

=> 9 ( y ) = 9 ( x + 3 )

=> y = x + 3

=> y - x = 3 ...(2)

$\star$ Now , solve (1) + (2) , we get ,

=> ( x + y ) + ( y - x ) = 7 + 3

=>  2 y = 10

=>  y = 5

$\star$ Substituting value of y in (1) , we get ,

=> x + (5) = 7

=> x = 7 - 5

=> x = 2

=> The Original number = 10 x + y = 10 (2) + 5

∴ The Original number = 20 + 5 = 25

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