Question

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

Answer 1

Given :

• Sum of the digits of a two digit number is 9.
• When we interchange the digit, the resulting new number is greater than the original number by 27.

To find :

• The original number.

Solution :

Let the digit at ten's place be x.

Let the digit at unit's place be y.

Original number = (10x + y)

Case 1 :

Sum of digits of a two digit number is 9.

Equation -

$\sf \implies x + y = 9 \\\\\implies \sf y = 9 - x\: \: \: \dots (i)$

Case 2 :

When we interchange the digit, the resulting new number is greater than the original number by 27.

Equation -

$\sf \implies 10y + x = 10x + y + 27 \\\\\implies \sf 10y - y + x - 10x = 27 \\\\\implies \sf 9y - 9x = 27 \\\\\implies \sf 9(y - x) = 27 \\\\\implies \sf y - x = \frac{27}{9} \\\\\implies \sf y- x = 3 \\\\\implies \sf 9-x-x = 3$

[ From equation (i) ]

$\sf \implies 9 - 2x = 3\\\\\implies \sf - 2x = 3-9 \\\\\implies \sf - 2x = - 6 \\\\\implies \sf x = \frac{-6}{-2} \\\\\implies \sf x = 3$

Substitute the value of x in equation (i),

$\sf \implies y = 9-x \\\\\implies \sf y = 9-3 \\\\\implies \sf y = 6$

$\large{\boxed{\tt Tens \: digit =x= 3}}$

$\large{\boxed{\tt Units \: digit = y=6}}$

$\large{\boxed{\tt Original \: Number = 10(x)+y=10(3)+6=30+6=36}}$

Answer 2

Answer:

$\bullet\large\underline\textsf{The\:two\:digit\:number\:is\:36}$

$\bullet\large\underline\textsf{One's\:place = 3}$

$\bullet\large\underline\textsf{Ten's\:place = 6}$

Step-by-step explanation:

$\large\underline\textsf{Given, }$

• Sum of two digit number = 9

$\large\underline\textsf{To\:Find, }$

• Two digit number = ?

$\large\underline\textsf{Solution\::}$

$\mapsto$Let the unit's digit to be x

$\mapsto$Let the ten's digit to be (9 - x)

$\mapsto$Now, let the original number :

10(9 - x) + x = 90 - 10x + x = 90 - 9x

$\mapsto$On interchanging the digits, the new number is :

10x + (9 - x) = 10x + 9 - x = 9x + 9

☯$\large\underline\textsf{According\:to\:the\:question}$

New number = Original number + 27

$\leadsto$9x + 9 = 90 - 9x + 27

$\leadsto$9x + 9 = 117 - 9x

$\leadsto$9x = 117 - 9 - 9x

$\leadsto$9x + 9x = 108

$\leadsto$18x = 108

$\leadsto$x = 108/18

$\bold\leadsto$x = 6

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So, now we will find out our original number/two digit number :

Original number = 90 - (9 × 6) = 90 - 54 = 36

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