Question

3. Sum of the digits of a two-digit number is 9. When we interchange the digits, it is
found that the resulting new number is greater than the original number by 27. What are the numbers
​

Answer 1

Let the two digits in number be x and y respectively .

• Then the number is 10x + y .

A/Q,

• sum of digits = 9

x + y = 9

y = 9 - x _____(1)

When the digits are interchanged , the ones digit becomes x and ten's digit becomes y .

• Hence , the new number is 10 y + x .

A/Q

• New number - 27 = original number

10 y + x - 27 = 10x + y

10 y - y + x - 10x - 27 = 0

9 y - 9x - 27 = 0

9 ( y - x - 3) = 0

y - x - 3 = 0/9

y - x - 3 = 0

y - x = 3

Substituting value of y from equation (1)

9 - x - x = 3

9 - 2x = 3

-2x = 3 - 9

-2x = -6

x = -6/-2

• x = 3
• x = 3 y = 9 - 3 = 6

Hence the original number is 36 .

New number is 63 .

Verification :

63 -27

= 36 (original number)

Hence verified

Answer 2

${\huge{\bf{\pink{Solution:-}}}}$

Given:-

⇢ Sum of the digits of a two-digit number is 9. When we interchange the digits, it is found that the resulting new number is greater than the original number by 27.

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Find:-

⇢ What are the numbers

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Calculations:-

• Let (x + y) be the two digit numbers.
• Let (x, y) be the original number.

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As per the first case, sum of two-digits number is 9.

⇢ $\sf{x + y = 9}$

⇢ $\sf{x = 10x + y}$

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Now, the resulting new number is greater than the original number is 27.

⇢ $\sf{(10y + x) - (10x + y) = 27}$ or

⇢ $\sf{9y - 9x = 27}$ or

⇢ $\sf{y - x = 3}$

⇢ $\bold{x = 3}$ㅤ

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⇢ $\sf{2y = 12}$ or

⇢ $\bold{y = 6}$

hence the required two digit original number is 36.

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VERIFICATION:-

⇢ $\sf{x + y = 9}$

⇢ $\sf{3 + 6 = 9}$

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HENCE PROVED!!!