Question

The sum of the digits of a two digir number is 7 . The number obtained by interchanging the digits exceeds the original number by 27 . Find the number

Answer 1

GIVEN:

• The sum of the digits of a two digit number is 7
• The number obtained by interchanging the digits exceeds the original number by 27

TO FIND:

• What are the numbers ?

SOLUTION:

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

 ❒ NUMBER = 10x + y

CASE:- 1

The sum of the digits of a two digit number is 7

According to question:-

$\bf{\hookrightarrow x + y = 7...1)}$

$\rm{\hookrightarrow x = 7-y }$

CASE:- 2

The number obtained by interchanging the digits exceeds the original number by 27.

• Number obtained by reversing the digits = 10y + x
• Number obtained by reversing the digits = Original number + 27

According to question:-

$\rm{\hookrightarrow 10y + x = 10x + y + 27 }$

$\rm{\hookrightarrow -27 = 10x+y-10y-x }$

$\rm{\hookrightarrow -27 = 9x-9y }$

Taking 9 as common from both sides

$\bf{\hookrightarrow -3 = x-y....2) }$

Put the value of 'x' in equation 2)

$\rm{\hookrightarrow -3 = 7-y-y }$

$\rm{\hookrightarrow -3-7 = -2y }$

$\rm{\hookrightarrow -10 = -2y }$

$\rm{\hookrightarrow \cancel\dfrac{-10}{-2} = y }$

$\bf{\hookrightarrow 5 = y }$

Put the value of 'y' in equation 1)

$\rm{\hookrightarrow x + 5 = 7 }$

$\rm{\hookrightarrow x = 7-5 }$

$\bf{\hookrightarrow x = 2 }$

• Number = 10x + y
• Number = 10(2)+5
• Number = 20+5
• Number = 25

❝ Hence, the number is 25 ❞

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