Question

a number consists of two digits whise sum is 9.If 27 is subtracted fron tge original number,its digits are interchanged.Then the original number is....​

Answer 1

Answer:

The Original Number is 63.

Step-by-step explanation:

Given :

Sum of the digits = 9

When 27 is subtracted from the original number = the digits get interchanged

To find :

The original number

Solution :

$\textsf{\underline{\underline{Original Number :-}}}$

Let the Digits be -

• Units Place as y
• Tens Place as 10(9 - y)

⇒ 10(9 - y) + y

⇒ 90 - 10y + y

⇒ 90 - 9y ........ [Original Number]

$\rule{300}{1.5}$

$\textsf{\underline{\underline{Number with Reversed Digits :- }}}$

Let the Digits be -

• Units Place as (9 - y)
• Tens Place as 10(y)

⇒ 10(y) + (9 - y)

⇒ 10y + 9 - y

⇒ 9y + 9 ...... [Number with Reversed Digits]

$\rule{300}{1.5}$

$\textsf{\underline{\underline{According to the Question :- }}}$

When 27 is subtracted from the original number, the digits get interchanged.

⇒ (90 - 9y) - 27 = 9y + 9

⇒ 63 - 9y = 9y + 9

⇒ 9y + 9y = 63 - 9

⇒ 18y = 54

⇒ y = $\tt{\dfrac{54}{18}}$

⇒ y = 3

$\rule{300}{1.5}$

★ Value of 90 - 9y (Original Number)

⇒ 90 - 9(3)

⇒ 90 - 27

⇒ 63

⇒ Original Number = 63

$\therefore$ The Original Number is 63.

$\rule{300}{1.5}$

Verification :

As the original number is 63, the number with interchanged digits will be 36. Check whether the condition given in the question matches or not.

Condition - When 27 is subtracted from the original number, the digits get interchanged.

⇒ 63 - 27

⇒ 36

$\therefore$ The Original Number is 63.

Answer 2

$\bold{\underline{\underline{Answer:}}}$

Original Number = 63

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

Given :

• A number consists of two digits whose sum is 9.
• If 27 is subtracted from the original number,its digits are interchanged.

To find :

• The original number

Solution :

Let the digit in the tens place be x.

Let the digit in the units place be y.

Original Number = 10x + y

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

• Sum of the digits = 9

Constituting it mathematically,

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

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

• 27 is subtracted from the original number,its digits are interchanged.

Reversed Number = 10y + x

Constituting it mathematically,

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

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

$\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 = 12

$\rightarrow$$\bold{x={\dfrac{12}{2}}}$

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

Substitute x = 6 in equation 1,

$\rightarrow$x + y = 9 ---> (1)

$\rightarrow$6 + y = 9

$\rightarrow$y = 9 - 6

$\rightarrow$y = 3

$\bold{\boxed{\red{\rm{Tens\:digit\:=\:x\:=\:6}}}}$

$\bold{\boxed{\red{\rm{Units\:digit\:=\:y\:=\:3}}}}$

$\bold{\boxed{\red{\rm{Original\:Number\:=\:10x+y\:=\:10\times\:6\:+\:3\:=\:60\:+\:3\:=\:63}}}}$