Question

A positive numver consists of two digits. If the digit in the unit place is divided by the digit in the tenth place then the quotient is 3. If 36 is added to the number then digits of the number are reversed. Find the number.

Answer 1

$\underline{\underline{\Huge{\textbf{Solution:}}}}$

Given,
For a Two-digit Number,

$\textit{Statement-1:}$
If the digit in the unit place is divided by the digit in the tenth place then the quotient is 3.

$\textit{Statement-2:}$
If 36 is added to the number then digits of the number are reversed.



$\underline{\large{\textbf{Step-1:}}}$
Rewrite Statement-1

Let the Unit's digit be x
Let the Ten's digit be y

We have,
$\bigstar \textbf{Dividend = Divisor \times Quotient + Remainder}$

Assuming that Remainder is 0.

$\implies x = 3y + 0$

$\implies x = 3y$ ————[1]



$\underline{\large{\textbf{Step-2:}}}$
Form the Number using the digits.

A Two digit No. is equal to the sum of 10 times it's Ten's digit and the Unit digit.

$\bigstar \textbf{Two-digit\: Number = 10(Ten's\: digit)+1(Unit's\: digit)}$

$\implies \textsf{Two- digit Number = 10(y)+x}$

Substituting [1]

$\implies$ Two- digit Number = 10(y)+(3y)

$\implies \textsf{Two- digit Number = 13y}$ ————[2]



$\underline{\large{\textbf{Step-3:}}}$
Find the Number when digits are reversed.

For a Two-digit Number, if the digits are reversed, the Unit's digit and Ten's digit are interchanged.

$\implies$ Reversed Two-digit Number = 10(3y)+y

$\implies \textsf{Reversed Two-digit Number = 31y}$ ————[3]



$\underline{\large{\textbf{Step-4:}}}$
Rewrite Statement-2

$\implies$ Original Number + 36 = Reversed Two-digit Number

$\implies \textsf{Reversed Two-digit Number - Original Number = 36}$

Substituting [2] & [3]

$\implies$ 31y-13y = 36

$\implies$ 18y = 36

$\implies$ y = 2 ————[4]



$\underline{\large{\textbf{Step-5:}}}$
Substitute [4] in [1]

$\implies$ x = 3(2)

$\implies$ x = 6 ————[5]



$\underline{\large{\textbf{Step-6:}}}$
Express the number using the digits.

Two-digit Number = 10(2)+6 = 26
Reversed Two-digit Number = 10(6)+2 = 62



$\therefore$
$\bigstar \textbf{The Original Two- Number = \underline{\large{26}} and Reversed Two-digit Number = \underline{\large{62}}}$