Question

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

Answer 1

Answer:

36

Step-by-step explanation:

number= 10x+y

x+y=9       ------------------------------------------- 1

interchange number= 10y+x

10y+x=10x+y +27

-9x+9y=27

-x+y=3      -------------------------------------------2

addition of eq1 and eq2

x+y=9

-x+y=3

-------------------------------------

2y= 12

y=6

sub y-=6 in eq 1

x+y=9

x+6=9

x=3

number = 10x+y= 10*3+6 =30+6 = 36

Answer 2

$\large\underline{\sf{Solution-}}$

Given that,

~ Sum of the digits of a two-digit number is 9.

$\begin{gathered}\begin{gathered}\bf\: Let-\begin{cases} &\sf{digit \: at \: tens \: place \: be \: x} \\ \\ &\sf{digits \: at \: ones \: place \: be \: 9 - x} \end{cases}\end{gathered}\end{gathered}$

So,

Original number = 10 × x + 1 × (9 - x) = 10x +9 - x = 9x + 9

Reverse number = 10 × (9 - x) + x × 1 = 90 - 10x + x = 90 - 9x

$\red{\begin{gathered}\begin{gathered}\bf\: So-\begin{cases} &\sf{number \: formed = 9x + 9} \\ \\ &\sf{reverse \: number = 90 - 9x} \end{cases}\end{gathered}\end{gathered}}$

According to statement,

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

$\red{\rm :\longmapsto\:Reverse \: number - Original \: number = 27}$

$\rm :\longmapsto\:90 - 9x - (9x + 9) = 27$

$\rm :\longmapsto\:90 - 9x - 9x - 9 = 27$

$\rm :\longmapsto\:81 - 18x = 27$

$\rm :\longmapsto\: - 18x = 27 - 81$

$\rm :\longmapsto\: - 18x = - 54$

$\bf\implies \:x = 3$

$\red{\begin{gathered}\begin{gathered}\bf\: So-\begin{cases} &\sf{number \: formed = 9x + 9 = 36} \\ \\ &\sf{reverse \: number = 90 - 9x = 63} \end{cases}\end{gathered}\end{gathered}}$

Thus,

• 2 digit number is 36