Question

The digit of a two digit number differ by 2. If the digit are interchanged of the resulting number is added to the original number, we get 154. Find the number.​

Answer 1

Solution :

$\bf{\blue{\underline{\underline{\bf{Given\::}}}}}$

The digit of a two digit number differ by 2. If the digit are interchanged of the resulting number is added to the original number, we get 154.

$\bf{\blue{\underline{\underline{\bf{To\:find\::}}}}}$

The number.

$\bf{\blue{\underline{\underline{\bf{Explanation\::}}}}}$

Let the ten's digit be r

Let the one's digit be (r+2)

$\bf{\underline{\sf{The\:Original\:number\:be=\red{10r+(r+2)}}}}\\\bf{\underline{\sf{The\:Reversed\:number\:be=\red{10(r+2)+r)}}}}$

A/q

$\mapsto\sf{10r+(r+2)+10(r+2)+r=154}\\\\\mapsto\sf{10r+r+2+10r+20+r=154}\\\\\mapsto\sf{11r+2+11r+20=154}\\\\\mapsto\sf{22r+22=154}\\\\\mapsto\sf{22r=154-22}\\\\\mapsto\sf{22r=132}\\\\\mapsto\sf{r=\cancel{\dfrac{132}{22} }}\\\\\mapsto\sf{\red{r=6}}$

Thus;

The original number = 10r+(r+2)

The original number = 10(6)+(6+2)

The original number = 60 + 8

The original number = 68.

The reversed number = 86

Answer 2

AnswEr:-

Original number = 68

$\rule{200}{1}$

Given:-

• Digits of a two digit number differs by 2

• Digits interchanged, added to original number,sum = 154

To find:-

• Original number = ?

Solution:-

• Let the digit at unit's place be x & digit at ten's place be y where x > y

• Therefore, original number = x + 10y

Case 1:-

⇒ x - y = 2 ____(Eq.1)

Case 2:-

⋆ Interchanged number:-

∴ New number = y + 10x

A/q,

⇒ (y + 10x) + (x + 10y) = 154

⇒ y + 10x + x + 10y = 154

⇒ 11x + 11y = 154

⇒ 11(x + y) = 154

⇒ x + y = 154/11

⇒ x + y = 14 _____(Eq.2)

★ Adding both equations:-

$\qquad\sf{x - \cancel{y} = 2}$

$\qquad\sf{x + \cancel{ y} = 14}$

$\quad\sf{(+) \: (+)\: (+)}$

$\: \: \rule{120}{1}$

$\qquad\sf{2x \: \: = \: \: 16}$

$\longrightarrow\sf{x = 16/2 }$

$\longrightarrow\sf x = \large{\boxed{\sf\green{8 }}}$

So, digit at unit's place is 8.

Put this value in (Eq.1)

⇒ 8 - y = 2

⇒ -y = 2 - 8

⇒ -y = -6

⇒ y = 6

So, digit at ten's place is 6 .

Now finding number:-

⇒ Original number = x + 10y

⇒ Original number = 8 + 10(6)

⇒ Original number = 8 + 60

⇒ Original number = 68

★ Interchanged number (If required!)-

⇒ Interchanged number = 6 + 10(8)

⇒ Interchanged number = 6 + 80

⇒ Interchanged number = 86

Therefore,

$\therefore\underline{\boxed{\textsf{Original number = {\textbf{68 }}}}}$