Question

A two-digit number is such that the product of digits is 20. If 9 is added to the number, the digits interchange their places. Find the number.

Answer 1

Answer:

Step-by-step explanation:

solution

 let the ones digit number be x+10y

 let the tens digit number be 10x+y

  product of number =x×y=20       equation 1

 9 is added to the number i.e original number

so

x+10y+9

now

x+10y+9=10x+y

10x-x+y-10y=9

9x-9y=9

9(x-y)=9

x-y=1

x=1+y       equation 2

Now

  substituting the value of x in equation 1

   x×y=20

 (1+y)×y=20

y+y²=20

y²+(5-4)y-20=0

y²+5y-4y-20=0

y(y+5)-4(y+5)=0

(y-4)(y+5)=0

Either

 y=4

Or

 y=-5

Now

  placing the value of y in equation 2

x=1+y

x=1+4

x=5

Now

  placing the value of x and y in the ones digit number i.e original number

x+10y

5+10×4

5+40

45

Now

   in a interchanged number i.e tens digit number

10x+y

10×5+4

50+4

54

         the numbers are 45 and 54

Answer 2

let tenth digit is X and unit digit is Y then the number is 10 X + Y

first case:-

xy=20.....I

second case :-

10x +y+9=10y+x

10x-x+y-10y=-9

9x-9y=-9

9(x-y)=-9

x-y=-1....ii

(i,ii):-

(x+y)2=(x-y)2 +4xy

(x+y)2=(-1)2+4*20

(x+y)2=1+80

(x+y)2=81

x+y=9....iii

then

x-y=-1....ii

x+y=9...iii

2x=10

x=5 ans.

then

y=x+1....(from eq. ii)

y=5+1

y=6ans.