Question

The sum of a two digit number and the number formed by interchanging the digits is 132. If 12 is added to the number, the new number becomes 5 times the sum of the digits. Find the number. ​

Answer 1

$\sf{\underline{\boxed{\huge{\blue{\mathsf{Solution}}}}}}$

$\sf{\bold{\green{\underline{\underline{Given}}}}}$

• Sum of 2 digit no. and the no. formed by interchanging the digits is 132
• If 12 is added to no. the new no. becomes 2 the sum of digits

_______________________

$\sf{\bold{\green{\underline{\underline{To\:Find}}}}}$

• Find the original number

______________________

$\sf{\bold{\green{\underline{\underline{Solution}}}}}$

• Let the original no. be 10x + y
• No. formed by interchanging the digits = 10y + x

ATQ

: $\sf\implies\: {\bold{ 10 x + y + 10y + x = 132}}$

⠀⠀

: $\sf\implies\: {\bold{ 11 x + 11y = 132}}$

⠀⠀⠀⠀

: $\sf\implies\: {\bold{ 11 ( x + y ) = 132}}$

⠀⠀⠀⠀

: $\sf\implies\: {\bold{ x + y = 12}}$------ ( i )

⠀⠀⠀⠀

ATQ to 2nd statement :-

: $\sf\implies\: {\bold{ 10x + y + 12 = 5 ( x + y )}}$

⠀⠀⠀⠀

: $\sf\implies\: {\bold{ 10x + y + 12 = 5x + 5y}}$

⠀⠀⠀⠀

: $\sf\implies\: {\bold{ 5y - y -10x + 5x = 12}}$

⠀⠀⠀⠀

: $\sf\implies\: {\bold{ 4y - 5x = 12}}$------ ( ii )

⠀⠀⠀⠀

• Multiply eq ( i ) by 5

⠀⠀⠀⠀

: $\sf\implies\: {\bold{ 5x - 5y = 60}}$----- ( iii )

⠀⠀⠀⠀

• Add eq ( ii ) and ( iii )

: $\sf\implies\: {\bold{ 4y - 5x + 5x + 5y = 12 + 60}}$

⠀⠀⠀⠀

: $\sf\implies\: {\bold{ 9y = 72}}$

⠀⠀⠀⠀

: $\sf\implies\: {\bold{ y = \dfrac{72}{9}}}$

⠀⠀⠀⠀

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

⠀⠀⠀⠀

$\sf{\red{\boxed{\bold{y = 8 }}}}$

⠀⠀

• Putting value of y in eq ( i )

: $\sf\implies\: {\bold{ x + y = 12}}$

⠀⠀⠀⠀

: $\sf\implies\: {\bold{ 8 + x = 12}}$

⠀⠀

: $\sf\implies\: {\bold{ x = 4}}$

⠀⠀

$\sf{\red{\boxed{\bold{x = 4}}}}$

⠀⠀⠀⠀

• Finding the digit

: $\sf\implies\: {\bold{ 10 x + y}}$

⠀⠀⠀⠀

: $\sf\implies\: {\bold{10\times 4 + 8}}$

⠀⠀⠀⠀

: $\sf\implies\: {\bold{ 40 + 8}}$

⠀⠀⠀⠀

: $\sf\implies\: {\bold{ 48 }}$

⠀⠀⠀⠀

$\sf{\red{\boxed{\bold{Required \: number = 48}}}}$

______________________

$\sf{\bold{\green{\underline{\underline{Answer}}}}}$

• Required number = 48

Answer 2

Answer:

48

Step by Step explanation:

Let the number be xy

Then the number is 10x+y

Interchanging the digits, we get yx

That is 10y+x

10x + y + 10y + x = 132

11x+11y = 132

11(X+y) = 132

x + y = 132/11

X + y = 12 (equation 1)

It is given that

10x+y+12 = 5(X+y)

10x+y+12 = 5x + 5y

5x-4y= -12 (equation 2)

(equation 1) is multiplied by 5

Then 5x+5y = 60

5x-4y = -12

Subtract equation 2 from equation 1

we get

9y=72

y = 8

X+y = 12

X+ 8 = 12

X = 12-8

X = 4

the number is 48

and the interchanged number is 84

so 48+84= 132...