Sum of the digits of a two - digit number is 9 . When we interchange the digits, it is found that the resulting interchange the digits, it is found that resulting number is greater than original number by 27. What us two digit number?

Given: Sum of the digits of a two - digit number is 9

When we interchange the digits, it is found that the resulting interchange the digits, it is found that resulting number is greater than original number by 27

To do: Find the two digit number

Solution:

Digit in one's place  =  $y$  

so, $1  \times  y  =  y$

Digit in ten's place  = $ x$

so, $10 \times x   =  10x$

The number is  $10 x  +  y$

Sum of the digits is 9

 $x  +  y  =  9$ .......................(i)

When we interchange the number,

$x$ will come to one's place

$1 \times x  =  x$

y will come to ten's place

$10 \times y  =  10 y$

So, the interchanged number is  $10 y + x$

Interchanged number is greater than actual number by 27

$(10 y + x)  -  (10 x + y)  =  27$

$10 y + x  - 10 x - y  =  27$    (multiply '-' inside the bracket) 

$10 y - y -10 x + x   =  27$

$9 y - 9 x  =  27$ .....................................(ii)

divide (ii)  by 9

$\frac{9 y}{ 9}   -  \frac{9 x }{ 9}  =  \frac{27}{ 9}$

$y  -  x  =  3$

rewrite,  $- x  + y  =  3 $ ......................(iii)

add  (i)  and   (iii)  equations,

$x  +  y  -  x  + y  =  9  + 3$              (x - x = 0)

 $2 y  =  12$

 $y   =  \frac{12}{ 2}$ 

$y  =  6$

Substitute $y =6$ in (i)

$x  +  6  = 9$

$x   =   9 - 6$

$x = 3$

The number in ones place is  $6  ;  1 \times 6 = 6$

The number in tens place is  $3  ;  10 \times 3 = 30$

So, the actual number is  $30+6  =  36$

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Updated on: 10-Oct-2022

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