Question

a 2 digit number becomes five-sixth of itself when it's digits are reversed.the two digits differ by one.find the number.
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Answer 1

Solution :

Let the digit at ten's place be x and the digit at unit's place be y

Difference between digits = 1

⇒ x - y = 1

⇒ x - 1 = y

Number formed by the digits = 10x + y

Number obtained when digits are reversed = 10y + x

Given :

Number becomes 5/6th itself when digits are reversed

$\implies \dfrac{5}{6} (10x + y) = 10y + x$

⇒ 5(10x + y) = 6(10y + x)

⇒ 50x + 5y = 60y + 6x

⇒ 50x - 6x = 60y - 5y

⇒ 44x = 55y

⇒ 4x = 5y

Substituting y = x - 1

⇒ 4x = 5(x - 1)

⇒ 4x = 5x - 5

⇒ 5 = 5x - 4x

⇒ 5 = x

⇒ x = 5

Substituting x = 5 in y = x - 1

⇒ y = 5 - 1 = 4

Number = 10x + y = 10(5) + 4 = 50 + 4 = 54

Hence, the number is 54.

Answer 2

Given :-----

• Two digits number becomes 5/6th of reversing number.
• Difference of digits = 1.

To Find :----

• Actual Number ?

Let the actual number unit digit = x

digit at 10th place = y ,

than our number will be = ( 10y +x )

Now, After reversing the digits ,

Digit at unit place = y

Digit at 10th place = x

so, new number will be = ( 10x + y)

now, it is given that, this new number is 5/6 of actual number .

so,

$\frac{5}{6} (10</strong><strong>y</strong><strong> + </strong><strong>x</strong><strong>) = (10</strong><strong>x</strong><strong> + </strong><strong>y</strong><strong>) \\ \\ 5(10</strong><strong>y</strong><strong> + </strong><strong>x</strong><strong>) = 6(10</strong><strong>x</strong><strong> + </strong><strong>y</strong><strong>) \\ \\ 50</strong><strong>y</strong><strong> + 5</strong><strong>x</strong><strong> = 60</strong><strong>x</strong><strong> + 6</strong><strong>y</strong><strong> \\ \\ 50</strong><strong>y</strong><strong> - 6</strong><strong>y</strong><strong> = 60</strong><strong>x</strong><strong> - 5</strong><strong>x</strong><strong> \\ \\ 44</strong><strong>y</strong><strong> = 55</strong><strong>x</strong><strong>$

44y - 55x = 0------------------------ Equation (1)

Now, Let y>x,

and , given,

y - x = 1 ------------------------ Equation(2)

Multiplying Equation (2) by 44 and subtracting from Equation (1) we get,

→ 44y - 55x - 44(y-x) = 0 - 44

→ 44y - 44y - 55x + 44x = (-44)

→ -11x = -44

→ x = (-44)/(-11) = 4

putting in Equation (2) we get,

→ y - 4 = 1

→ y = 5 ..

Hence , our Actual Two digits Number will be = (10y+x) = (10×5 + 4) = 54 (Ans)

(Hope it Helps you)