Question

The 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:

Step-by-step explanation:

Given :-

The 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.

To Find :-

The Two-digit Number

Solution :-

let the digits be x and y

Number = (10x + y)

interchanged digits = (10y + x)

According to the Question,

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

10y + x - (10x + y) = 27

10y + x - 10x -y =27

9y - 9x =27

9(y - x) = 27

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

Adding (i) and (ii), we get

y + x + y - x

= 9 + 3

= 12

⇒ 2y = 12  

⇒ y = 6

⇒ x = 3

Hence, The number is 36.

Answer 2

$\bold{\huge{\underline{\underline{\rm{AnsWer:}}}}}$

Original two digit number = 36.

$\bold{\large{\underline{\underline{\sf{Step\:by\:step\:explanation:}}}}}$

GIVEN :

• The sum of the digits of a two-digit number is 9.
• On interchanging the digits, it is found that the resulting new number is greater than the original number by 27

TO FIND :

• The two digit number.

SOLUTION :

Let the digit in the tens place be x.

Let the digit in the units place be y.

•°• Original two digit number = 10x + y

$\bold{\underline{\underline{\rm{\pink{As\:per\:the\:first\:given\:condition:}}}}}$

• Sum of the digits = 9

Constituting it mathematically,

$\longrightarrow$ $\bold{x+y=9}$ ---> (1)

$\bold{\underline{\underline{\rm{\pink{As\:per\:the\:second\:given\:condition:}}}}}$

• After interchanging the digits, the resulting new number is greater than the original two digit number by 27.

Two digit number after interchanging the digits will be

• 10y + x

Constituting it mathematically,

$\rightarrow$ $\bold{10y+x\:=\:10x+y\:+27}$

$\rightarrow$ $\bold{10y-y\:=\:10x\:-\:x\:+27}$

$\rightarrow$ $\bold{9y\:=\:9x\:+\:27}$

$\rightarrow$ $\bold{9x+27=9y}$

$\rightarrow$ $\bold{9x-9y\:=\:-27}$

$\rightarrow$ $\bold{9(x-y)\:=\:-27}$

$\rightarrow$ $\bold{x-y\:=\:{\dfrac{-27}{9}}}$

$\rightarrow$ $\bold{x-y\:=\:-3}$ --->(2)

Solve equation 1 and equation 2 simultaneously by elimination method.

Add equation 1 to equation 2,

x + y = 9 ----> (1)

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

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

2x = 6

$\rightarrow$ $\bold{x\:=\:{\dfrac{6}{2}}}$

$\rightarrow$ $\bold{x=3}$

Substitute x = 3 in equation 2,

$\rightarrow$ $\bold{x-y=-3}$

$\rightarrow$ $\bold{3-y=-3}$

$\rightarrow$ $\bold{-y\:=\:-3\:-\:3}$

$\rightarrow$ $\bold{-y=-6}$

$\rightarrow$ $\bold{y=6}$

$\bold{\large{\boxed{\sf{\purple{Tens\:digit\:=\:x\:=\:3}}}}}$

$\bold{\large{\boxed{\sf{\purple{Units\:digit\:=\:y\:=\:6}}}}}$

$\bold{\large{\boxed{\sf{\purple{Original\:Number\:=\:10x+y\:=\:10\times\:3\:+\:6\:=30\:+\:6\:=\:36}}}}}$