Question

the sum of the two digit number is 12. if the new no. formes by reversing the digit is greater than the originol no. by 54. find the originol no.​

Answer 1

$\mathfrak{\large{\underline{\underline{Answer :-}}}}$

The Original Number is 39

$\mathfrak{\large{\underline{\underline{Explanation :-}}}}$

Given :

Sum of the Digits = 12

The number with interchanged digits is greater than the original number by 54.

To Find :

The Number

Solution :

$\textbf{\small{\underline{Consider the Digits of Original Number as - }}}$

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

$\rightarrow$ x + 10(12 - x)

$\rightarrow$ x + 120 - 10x

$\rightarrow$ -9x + 120 ......... [ Original Number ]

$\rule{300}{1.5}$

$\textbf{\small{\underline{The Digits of the Reversed Number -}}}$

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

$\rightarrow$ 12 - x + 10x

$\rightarrow$ 12 + 9x .......... [ Number with Reversed Digits ]

$\rule{300}{1.5}$

$\textbf{\small{\underline{According to the Question - }}}$

The number with interchanged digits is greater than the original number by 54.

★ $\boxed{\tt{12+9x=(-9x + 120)+54}}$

$\tt{\longrightarrow} \: 12+9x=(-9x + 120)+54$

$\tt{\longrightarrow} \:12 + 9x = - 9x + 174$

$\tt{\longrightarrow} \:9x + 9x = 174 - 12$

$\tt{\longrightarrow} \:18x = 162$

$\tt{\longrightarrow} \:x = \dfrac{162}{18}$

$\tt{\longrightarrow} \:x = 9$

Units Place = 9

$\rule{300}{1.5}$

★ Value of 10(12 - x)

$\tt{\longrightarrow} \:10(12 - 9)$

$\tt{\longrightarrow} \: 10(3)$

$\tt{\longrightarrow} \:30$

$\rule{300}{1.5}$

The Original Number -

$\tt{\longrightarrow} \: 30 + 9$

$\tt{\longrightarrow} \: 39$

$\therefore$ The Original Number is 39.

$\rule{300}{1.5}$

$\mathfrak{\large{\underline{\underline{Verification :-}}}}$

Reversed digits of 39 = 93

Check if 93 is more 39 by 54 or not.

$\tt{\longrightarrow} \: 93-39$

$\tt{\longrightarrow} \: 54$

$\therefore$ The Original Number is 39.

Answer 2

» The sum of the two digit number is 12.

• Let one's digit be M and ten's digit be N.

• Original number = 10N + M

A.T.Q.

→ M + N = 12

→ M = 12 - N _________ (eq 1)

______________________________

» The new number formes by reversing the digit is greater than the originol number by 54.

• Reversed number = 10M + N

A.T.Q.

→ 10M + N = 10N + M + 54

→ 10M - M + N - 10N = 54

→ 9M - 9N = 54

→ M - N = 6

→ (12 - N) - N = 6 [From (eq 1)]

→ 12 - N - N = 6

→ - 2N = 6 - 12

→ - 2N = - 6

→ N = 3

Put value of N in (eq 1)

→ M = 12 - 3

→ M = 9

_____________________________

From above calculations we get M = 9 and N = 3

We have to find the number i.e 10N + M

Put value of N and M in it

→ 10(3) + 9

→ 30 + 9

→ 39

______________________________

39 is the original number.

_______ [ ANSWER ]

______________________________

✡ VERIFICATION :

From above calculations we get N = 3 and M = 9

Put them in this equation : 10M + N = 10N + M + 54

=> 9M - 9N = 54

=> 9(9) - 9(3) = 54

=> 81 - 27 = 54

=> 54 = 54

_____________________________