Question

. A number consists of two digits, where the number is divided by the sum of its digits,

the quotient is 7, if 27 is subtracted from the number, the digits interchange their places,

find the number.​

Answer 1

Step-by-step explanation:

Given:-

• A two digit number when divided by the sum of its digits gives the quotient 7.

• If 27 is subtracted from the number, the digits are interchanged.

To Find:-

• The original number.

Solution:-

Let the tens digit of the number be x

And, units digit of the number be y

Then,

The original number = 10x + y

The number obtained by interchanging the digits = 10y + x

Case 1:-

$\rm \dashrightarrow\dfrac{The \: original \: number}{Sum \: of \: its \: digits} = 7$

$\rm \dashrightarrow\dfrac{10x + y}{x + y} = 7$

$\rm \dashrightarrow10x + y = 7(x + y)$

$\rm \dashrightarrow10x + y = 7x + 7y$

$\rm \dashrightarrow10x + y - 7x - 7y = 0$

$\rm \dashrightarrow3x - 6y= 0$

Dividing the whole equation by 3

$\rm \dashrightarrow x - 2y= 0.....(i)$

Case 2:-

$\rm \dashrightarrow Original \: number- 27 = Reversed \: number$

$\rm \dashrightarrow 10x + y - 27 = 10y + x$

$\rm \dashrightarrow 10x + y - 10y - x = 27$

$\rm \dashrightarrow 9x - 9y = 27$

Dividing the whole equation by 9

$\rm \dashrightarrow x - y = 3......(ii)$

Subtracting equation (i) from (ii)

$\rm \dashrightarrow x - y - (x - 2y) = 3 -0$

$\rm \dashrightarrow x - y - x + 2y= 3$

$\rm \dashrightarrow y = 3$

Substituting y = 3 in equation (ii)

$\rm \dashrightarrow x - y = 3$

$\rm \dashrightarrow x - 3 = 3$

$\rm \dashrightarrow x = 3 + 3$

$\rm \dashrightarrow x = 6$

We have:-

• x = 6

• y = 3

The original number = 10x + y = 10(6) + 3 = 63

$\underline{\boxed{\purple{\rm \therefore The \: original \: number = 63}}}$