Question

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

Answer 1

Solution :

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

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

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

The number.

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

Let the ones place digit be r

Let the tens place digit be m

$\underline{\sf{The\:original\:number=\pink{10m+r}}}\\\underline{\sf{The\:reversed\:number=\pink{10r+m}}}$

A/q

$\leadsto\sf{r+m=9}\\\\\leadsto\sf{r=9-m..........................(1)}$

&

$\longrightarrow\sf{[10r+m]-[10m+r]=27}\\\\\\\longrightarrow\sf{10r+m-10m-r=27}\\\\\\\longrightarrow\sf{9r-9m=27}\\\\\\\longrightarrow\sf{9(r-m)=27}\\\\\\\longrightarrow\sf{r-m=\cancel{\dfrac{27}{9} }}\\\\\\\longrightarrow\sf{r-m=3}\\\\\\\longrightarrow\sf{9-m-m=3}\\\\\\\longrightarrow\sf{9-2m=3}\\\\\\\longrightarrow\sf{-2m=3-9}\\\\\\\longrightarrow\sf{-2m=-6}\\\\\\\longrightarrow\sf{m=\cancel{\dfrac{-6}{-2} }}\\\\\\\longrightarrow\sf{\green{m=3}}$

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

$\longrightarrow\sf{r=9-3}\\\\\longrightarrow\sf{\green{r=6}}$

Thus;

$\underbrace{\bf{The\;original\:number\:10m+r=10(3)+6=30+6=\boxed{36}}}}}$

Answer 2

Given:

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

To find

two-digit number .

Solution:

Let ten's digit number = x

and one's digit number = x

original no= 10x+ y

New number= 10y+x

sum of digits = x+y = 9............(1)

New number = original number+27

→ 10y+x = 10x+y+27

→10y-y+x-10x=27

→9y-9x=27

→y-x=27/9

→y-x=3.........(2)

from (1) and (2)

x+y = 9........(1)

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

by substitution method,

x = 9-y ......(3)

put value of (3) in equation (2)

→y-(9-y) = 3

→y-9+y = 3

→2y= 9-3

→2y= 6

→y = 6/2

→y = 3

Now put the value of y in equation (3)

→x = 9-3

→x = 6

therefore the value of x = 6 and y = 3

So the number is 36