A fraction becomes 13, if 1 is subtracted from both its numerator and denominator. If 3 is added to both the numerator and the denominator, it becomes 12. Find the fraction.
Let the numerator be x and the denominator be y.
From the given information, we have,
x−1y−1=13andx+3y+3=12or,3(x−1)=1(y−1)=>3x−3=y−1=>3x−y−2=0Also,2(x+3)=1(y+3)=>2x+6=y+3=>2x−y+3=0\\
Using cross-multiplication method,
x(−1)(3)−(−1)(−2)=y(−2)(2)−(3)(3)=1(3)(−1)−(2)(−1)x−3−2=y−4−9=1−3+2x−5=y−13=1−1=>x−5=1−1or,x=5=>y−13=1−1or,y=13
Thus, the original fraction is xy=513