Question

The Two digit number is obtained by either multipling sum of digit by 8 and adding 1 or by multiplying the difference of of the digit by 13 and adding 2 find the number
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Answer 1

$\huge\purple{\underline{\underline{\pink{Ans}\red{wer:-}}}}$

$\sf{The \ number \ is \ 41.}$

$\sf\orange{Given:}$

$\sf{\implies{The \ two \ digit \ number \ is}}$

$\sf{obtained \ by \ either \ multiplying \ sum}$

$\sf{of \ digit \ by \ 8 \ and \ adding \ 1.}$

$\sf{\implies{By \ multiplying \ the \ difference}}$

$\sf{of \ the \ digit \ by \ 13 \ and \ adding \ 2.}$

$\sf\pink{To \ find:}$

$\sf{The \ number.}$

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

$\sf{Let \ the \ digit \ in \ the \ ten's \ place \ be}$

$\sf{x \ and \ in \ unit's \ place \ be \ y.}$

$\sf{\therefore{Number=10x+y}}$

$\sf{According \ to \ first \ condition.}$

$\sf{10x+y=8(x+y)+1}$

$\sf{10x+y=8x+8y+1}$

$\sf{10x-8x+y-8y=1}$

$\sf{2x-7y=1...(1)}$

$\sf{According \ to \ second \ condition.}$

$\sf{10x+y=13(x-y)+2}$

$\sf{10x+y=13x-13y+2}$

$\sf{10x-13x+y+13y=2}$

$\sf{-3x+14y=2...(2)}$

$\sf{Multiply \ eq(1) \ by \ 2 \ we \ get,}$

$\sf{4x-14y=2...(3)}$

$\sf{Add \ equations \ (3) \ and \ (2) \ we \ get,}$

$\sf{-3x+14y=2}$

$\sf{+}$

$\sf{4x-14y=2}$

_______________________

$\sf{\therefore{x=4}}$

$\sf{Substitute \ x=4 \ in \ eq(1) \ we \ get,}$

$\sf{2(4)-7y=1}$

$\sf{8-7y=1}$

$\sf{-7y=1-8}$

$\sf{-7y=-7}$

$\sf{y=\frac{-7}{-7}}$

$\sf{\therefore{y=1}}$

$\sf{\therefore{The \ number=10x+y}}$

$\sf{=10(4)+1}$

$\sf{=40+1}$

$\sf{=41}$

$\sf\purple{\tt{\therefore{The \ number \ is \ 41.}}}$

Answer 2

$\large{\underline{\bf{\purple{Given:-}}}}$

• The Two digit number is obtained by either multipling sum of digit by 8 and adding 1

• or by multiplying the difference of of the digit by 13 and adding 2.

$\large{\underline{\bf{\purple{To\:Find:-}}}}$

• we need to find the number.

$\huge{\underline{\bf{\red{Solution:-}}}}$

• Let the ten's place digit be x
• Let the unit's place digit be y

So,

• The Number = 10x + y

Condition I :-

The number is obtained if multiplying sum of digits by 8 and add 1.

⠀⠀⠀⠀⠀➝ 8(x +y ) + 1 = 10x +y

⠀⠀⠀⠀⠀➝ 8x + 8y +1 = 10x + y

⠀⠀⠀⠀⠀➝ 10x - 8x +y - 8y = 1

⠀⠀⠀⠀⠀➝ 2x - 7y = 1 ............(i)

Condition II :-

The number is obtained by multiplying the difference of of the digit by 13 and adding 2

⠀⠀⠀⠀⠀➝ 13(x - y) +2 = 10x +y

⠀⠀⠀⠀⠀➝ 13x - 13y + 2 = 10x +y

⠀⠀⠀⠀⠀➝ 13x - 10x - 13y - y = -2

⠀⠀⠀⠀⠀➝ 3x - 14y = -2 ............(ii)

• Multiplying equation (i) by 2

⠀⠀⠀⠀⠀➝ (2x - 7y =1) ×2

⠀⠀⠀⠀⠀➝ 4x - 14y = 2 ...........(iii)

• From eq. (ii) and (iii)

⠀⠀⠀⠀⠀4x - 14y = 2

⠀⠀⠀⠀⠀3x - 14y = -2

⠀⠀⠀⠀⠀--⠀⠀+⠀⠀⠀+

⠀⠀⠀⠀⠀x ⠀⠀⠀= 4

Now putting value of x in equation (ii)

⠀⠀⠀⠀⠀➝ 3x - 14y = -2

⠀⠀⠀⠀⠀➝3 ×4 - 14y = -2

⠀⠀⠀⠀⠀➝ 12 - 14y = -2

⠀⠀⠀⠀⠀➝ - 14y = -2 -12

⠀⠀⠀⠀⠀➝ - 14y = -14

⠀⠀⠀⠀⠀➝ y = -14/-14

⠀⠀⠀⠀⠀➝ y = 1

so,

The number(10x + y ) is = 10 ×4 +1

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀= 40 +1

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀= 41

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