Question

23. The sum of the digits of a two digit number is 12. The number obtained by interchanging the
digits is greater than the original number by 54. Find the original number.​

Answer 1

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

Let ,

The digit in ones place = x

So, the digit in tens place = 12 - x

Original no. = 10(12 - x) + 1(x)

= 120 - 10x + x

= 120 - 9x

New no. = 10(x) + 1(12 - x)

= 10x + 12 - x

= 9x + 12 [∵By reversing the digits]

According to Question,

⇒ New no. - Original no. = 54

⇒ (9x + 12) - (120 - 9x) = 54

⇒ 9x + 12 - 120 + 9x = 54

⇒ 18x - 108 = 54

⇒ 18x = 54 + 108

⇒ 18x = 162

⇒ x = 162 / 18

⇒ x = 9

Therefore ,

Original no. = 120 - 9x = 120 - 9(9)

= 120 - 81 = 39

New no. = 93 [∵ By reversing the digits]

Hence, the required number is either 39 or 93

Answer 2

Answer:

The Original Number is 39.

Step-by-step explanation:

Given :

Sum of the digits = 12

The number obtained by interchanging the

digits is greater than the original number by 54.

To find :

The Original Number

Solution :

Original Number -

$\textbf{\small{\underline{Let the digits be - }}}$

• Units Place as x
• Tens place as 10(12 - x)

⇒ 10(12 - x) + x

⇒ 120 - 10x + x

⇒ 120 - 9x ......... [Original number]

$\rule{300}{1.5}$

Number with reversed digits -

$\textbf{\small{\underline{Let the digits be -}}}$

• Units Place as (12 - x)
• Tens Place as 10(x)

⇒ 10(x) + (12 - x)

⇒ 10x + 12 - x

⇒ 9x + 12 ......... [Number with reversed digits]

$\rule{300}{1.5}$

According to the Question -

The number obtained by interchanging the

digits is greater than the original number by 54.

⇒ (120 - 9x) + 54 = 9x + 12

⇒ 174 - 9x = 9x + 12

⇒ 9x + 9x = 174 - 12

⇒ 18x = 162

⇒ x = 162/8

⇒ x = 9

$\rule{300}{1.5}$

★ Original Number -

⇒ 120 - 9(9)

⇒ 120 - 81

⇒ 39

Original number = 39

$\therefore$ The Original Number is 39.