Question

the sum of the digits of a two digit number is 9.If 9 is added to the numberby reversing its digits,then the result is thrice the original number.find the original number ​

Answer 1

$\huge\mathfrak\blue{Answer:}$

Given:

• We have been given a two digit number having 9 as the sum of its digits
• When 9 is added to number by reversing the digit the result is thrice the original number

To Find:

• We have to find the number

Solution:

Let the digit at Tens place be = x

Digit at One's place = y

$\boxed{\sf{\green{Original \: Number = 10x + y}}}$

$\boxed{\sf{\green{Reversed \: Number = 10y + x }}}$

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$\bigstar \: \: \underline{\large\mathfrak\orange{According \: to \: first \: Condition:}}$

$\hookrightarrow \sf{Sum \: of \: digits = 9 }$

$\hookrightarrow \sf{ x + y = 9 }$

$\hookrightarrow \boxed{\sf{ y = 9 - x }}$ ----------- ( 1 )

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$\bigstar \: \: \underline{\large\mathfrak\orange{According \: to \: second \: Condition:}}$

$\hookrightarrow \sf{(Reversed \: Number)+9 = 3(Original \: Number)}$

$\hookrightarrow \sf{ (10y+x) + 9 = 3( 10x+y )}$

$\hookrightarrow \sf{ 10y+x + 9 = 30x + 3y}$

Separating like terms on same side of equation

$\hookrightarrow \sf{29x - 7y = 9}$

Putting value of x from equation ( 1 )

$\hookrightarrow \sf{29(9-y) - 7y = 9 }$

$\hookrightarrow \sf{261 - 29y - 7y = 9 }$

$\hookrightarrow \sf{36y = 252 }$

$\hookrightarrow \sf{y = \dfrac{252}{36} }$

$\hookrightarrow \sf{y = \cancel{\dfrac{252}{36} }}$

$\hookrightarrow \sf{y = 7 }$

Putting value of y in Equation ( 1 )

$\longrightarrow \sf{ x = 9 - y }$

$\longrightarrow \sf{ x = 9 - 7 }$

$\longrightarrow \sf{ x = 2 }$

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Hence original number is

$\implies \sf{Original \: Number = 10x + y}$

$\implies \sf{Original \: Number = 10(2) + 7}$

$\implies \sf{Original \: Number = 20+ 7}$

$\implies \sf{Original \: Number = 27}$

$\boxed{\large\mathfrak\red{Original \: Number = 27 }}$

Answer 2

Let the original number be = 10x + y

Then x + y = 9

=> y = 9 - x ___(1)

According to the question:

10y + x + 9 = 3(10x + y)

=> 10y + x + 9 = 30x + 3y

=> 7y + 9 = 29x

Put (1):

=> 7(9 - x) + 9 = 29x

=> 63 - 7x + 9 = 29x

=> 72 = 36x

=> x = 2

y = 9 - x

=> y = 7

Number: 27