Does the value of a fraction change if you add the same number to the numerator and the denominator?

Answer 1

Yes it does. Don't do it!!!

You can see a fraction as an instruction: "Divide the top by what is on the bottom." If you divide it by 2, you double the bottom. If you multiply by 2 you double the top. So doubling the top and bottom is the same as multiplying by 2 then dividing by 2, which is the same as multiplying by 1, which is the multiplicative identity, so its the same as doing nothing. This effect is called Associativity: Multiplying one part by 2 then dividing another part by 2 when they are both going to be multiplied or divided is the same as multiplying by 1. However, this only works as division is a special case of multiplication: The multiplication by 1/something .

Addition is NOT associative. This property only works with multiplication. Adding something to top and bottom changes the value. (It doesn't add the value you added to the total of the fraction either; to do that you need to do fraction addition with the added fraction being the (addend*LCM)/(LCM) + ((original numerator * LCM)/original denominator)/(LCM) after finding the LCM - least common multiple)

A simple example of the fact it changes the value is the equation x/y=1: if you claim (x+1)/(y+1)=1, multiply both sides by y+1 to get x+1=y+1. Now subtract y from both sides to get x-y+1=1. According to this x-y+1=x/y (i.e. in all cases and values of x and y). This is clearly untrue, so the point has been made. (For any people who are into such things, I apologize for that truly horribly lax proof I just did.)