Question

b) A two digit number is obtained by multiplying the sum of the digits by 8.Also, it is obtained by multiplying the difference of the digits by 14 and adding 2. Find the number​

Answer 1

Let two digit number be 10x+y

Given 10x+y=(x+y)8−−−−−−−−(1)

⟹14∣x−y∣+2=100x+y, let x>y

⟹14(x−y)+2=10x+y

⟹4x−15y+2=0−−−−−−−−(2)

from (1) 2x=7y−−−−−−−−(3)

From (2) & (3)

14y−15y+2=0

y=2

When y=2,x=7

So number is 72.

Answer 2

$\large\underline{\sf{Solution-}}$

$\begin{gathered}\begin{gathered}\bf\: Let-\begin{cases} &\sf{digit \: at \: tens \: place \: be \: x} \\ \\ &\sf{digits \: at \: ones \: place \: be \: y} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf\: So-\begin{cases} &\sf{number \: formed = 10x + y} \\ \\ &\sf{reverse \: number = 10y + x} \end{cases}\end{gathered}\end{gathered}$

According to first condition

A two digit number is obtained by multiplying the sum of the digits by 8.

$\rm :\longmapsto\:10x + y = 8(x + y)$

$\rm :\longmapsto\:10x + y = 8x + 8y$

$\rm :\longmapsto\:10x - 8x = 8y - y$

$\rm :\longmapsto\:2x = 7y - - - - (1)$

According to second condition

A two digit number is obtained by multiplying the difference of the digits by 14 and adding 2

$\rm :\longmapsto\:10x + y = 14(x - y) + 2$

OR

$\red{\rm :\longmapsto\:10x + y = 14(y - x) + 2}$

$\rm :\longmapsto\:10x + y = 14x - 14y + 2$

OR

$\red{\rm :\longmapsto\:10x + y = 14y - 14x + 2}$

$\rm :\longmapsto\:10x + y - 14x + 14y = 2$

OR

$\red{\rm :\longmapsto\:10x + y - 14y + 14x = 2}$

$\rm :\longmapsto\: 15y - 4x= 2$

OR

$\red{\rm :\longmapsto\:24x- 13y= 2}$

$\rm :\longmapsto\:15y - 2(2x) = 2$

OR

$\red{\rm :\longmapsto\:12(2x)- 13y= 2}$

On substituting the value of 2x from equation (1), we get

$\rm :\longmapsto\:15y - 2(7y) = 2$

OR

$\red{\rm :\longmapsto\:12(7y)- 13y= 2}$

$\rm :\longmapsto\:15y - 14y = 2$

OR

$\red{\rm :\longmapsto\:84y - 13y = 2}$

$\rm :\longmapsto\:y = 2$

OR

$\red{\rm :\longmapsto\:71y = 2\rm \implies\:y = \dfrac{2}{71} \: \: which \: is \: not \: possible}$

Thus,

$\rm \implies\:\boxed{ \tt{ \: y \: = \: 2 \: }}$

On substituting the value of y in equation (1), we get

$\rm :\longmapsto\:2x = 7 \times 2$

$\rm \implies\:\boxed{ \tt{ \: x \: = \: 7 \: }}$

$\begin{gathered}\begin{gathered}\bf\: Hence-\begin{cases} &\sf{digit \: at \: tens \: place \: be \: 7} \\ \\ &\sf{digits \: at \: ones \: place \: be \: 2} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf\: So-\begin{cases} &\sf{number \: formed = 10x + y = 10(7) + 2 = 72} \\ \\ &\sf{reverse \: number = 10y + x = 10(2) + 7 = 27} \end{cases}\end{gathered}\end{gathered}$

• Hence, two digit number is 72.