A fraction becomes 911 if 2 is added to both the numerator and the denominator. If 3 is added to both the numerator and the denominator, it becomes 56. Find the fraction.
Let the fraction be xy, y≠0
Case1:
fraction becomes 911 if 2 is added to both the numerator and the denominator.
⇒x+2y+2=911
⇒11(x+2)=9(y+2)
⇒11x+22=9y+18
⇒11x−9y+22−18=0
⇒11x−9y+4=0 ---(1)
Case2:
fraction becomes 56 if 3 is added to both the numerator and the denominator.
⇒x+3y+3=56
⇒6(x+3)=5(y+3)
⇒6x+18=5y+15
⇒6x−5y+18−15=0
⇒16x−5y+3=0 ---(2)
Lets solve (1) and (2)
Multiply 11x−9y+4=0 by 5
⇒55x−45y+20=0----(3)
Multiply 6x−5y+3=0 by 9
⇒54x−45y+27=0 ---(4)
Lets do (3)-(4)
⇒55x−45y+20−(54x−45y+27)=0
⇒55x−45y+20−54x+45y−27)=0
⇒x−7=0
⇒x=7
Substitute x=7 in 11x−9y+4=0
⇒11(7)−9y+4=0
⇒77−9y+4=0
⇒81−9y=0
⇒81=9y
⇒819=y
⇒9=y
∴ The required fraction is 79.