A fraction becomes 4/7 when 1 is subtracted from its numerator, and it becomes 1/2 when 4 is added to its denominator. Find the fraction.

Let the fraction be $x/y.(x-1)/y=4/7$ gives $7x-4y=7$, and $x/(y+4)=1/2$ gives $2x-y=4$, so $y=2x-4$. Substituting: $7x-8x+16=7$, so $x=9$ and $y=14$. Check: 8/14 = 4/7 and 9/18 = 1/2. 8/14 is the fraction after 1 is taken off the numerator; 9/18 is the fraction after 4 is added to the denominator; 14/9 has the numerator and denominator swapped.