Question

The sum of the digits of a two digit number is 9 . Also nine this number is twice the number obtained by reversing the order of the digits. Find the number​

Answer 1

ANSWER:

• Original number = 18

GIVEN:

• sum of the digits of a two digit number is 9 .
• Nine times this number is twice the number obtained by reversing the order of the digits.

TO FIND:

Original number

SOLUTION:

• Let the digit at tens place be 'x'
• Let the digit at once place be 'y'

Original number= 10x+y

Case 1

$\implies \: x + y = 9 \: \: ...(i)$

Case 2

REVERSED NUMBER = 10y+x

$\implies \: 9(10x +y ) = 2(10y + x) \\ \implies \: 90x + 9y = 20y + 2x \\ \implies \: 90x - 2x = 20y - 9y \\ \implies \: 88x = 11y \\ \implies \: \frac{88x}{11} = y \\ \implies \: 8x = y \: \: \: ......(ii)$

Putting : y = 8x in eq (i)

$\implies \: x + 8x = 9 \\ \implies \: 9x = 9 \\ \implies \: x = 1$

Putting x= 1 in eq(ii)

$\implies \: y = 8 \times 1 \\ \implies \: y = 8$

ORIGINAL NUMBER = 10x+y

= 10*1 +8

= 18

Answer 2

Given :

• The sum of the digits of a two digit number is 9.
• Also nine *times* this number is twice the number obtained by reversing the order of the digits.

To Find :

• The original two digit number.

Solution :

Let the digit at the tens place be x.

Let the digit at the units place be y.

Original Number = 10x+y

Case 1 :

The sum of the ten's digit and units digit is 9.

Equation :

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

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

Case 2 :

9 times the original number is twice the number obtained by reversing the digits order.

Reversed Number = 10y + x

Equation :

$\implies$ $\sf{9(10x+y)=2(10y+x)}$

$\implies$ $\sf{90x+9y=20y+2x}$

$\implies$ $\sf{90x-2x=20y-9y}$

$\implies$ $\sf{88x=11y}$

$\implies$$\sf{88(9-y) =11y}$

$\bold{\big[From\:equation\:(1)\:x\:=\:9-y\big]}$

$\implies$ $\sf{792-88y=11y}$

$\implies$ $\sf{792=11y+88y}$

$\implies$ $\sf{792=99y}$

$\implies$ $\sf{\dfrac{792}{99}=y}$

$\implies$ $\sf{y=8}$

Substitute, y = 8 in equation (1),

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

$\implies$ $\sf{x=9-8}$

$\implies$ $\sf{x=1}$

$\large{\boxed{\bold{Ten's\:digit\:=\:x\:=\:1}}}$

$\large{\boxed{\bold{Unit's\:digit\:=\:y\:=\:8}}}$

$\large{\boxed{\bold{\purple{Original\:Number\:=\:(10x+y)=10(1)+8=10+8=18}}}}$