Proof from adding and subtracting fractions

Question

In adding or subtracting fractions with two or more terms, the resulting denominator is always the product of the denominators of the terms.

Can someone prove or disprove this statement. I'm pretty sure this isn't true, but I don't know how to disprove it. Does a statement like 1/2+1/3+2/3 count?

Answer 1

It is true if you consider fractions other than the lowest term (or simplest form).

Consider $\frac ab+\frac cd=\frac{ad+bc}{bd}$. Similarly for subtraction.

Clearly, if you reduce to the lowest term, it isn't true. Simply consider something like $\frac 14+\frac 14=\frac 12$; 2 isn't a product of 4 and 4.

Answer 2

It's not always true. When we add or subtract two or more fractions then resulting denominator is the LCM (Least Common Denominator) of the denominators of the terms, being added or subtracted. For example :

$\dfrac{1}{3}+\dfrac{1}{4}=\dfrac{7}{12}$ here $12$ is the LCM of 3 and 4.

Since the LCM of two or more prime numbers is their product, therefore if two or more fractions which have prime numbers as their denominator are added or subtracted then the resulting denominator will be their product, as we show in above examle.

Consider another example :

$\dfrac{1}{2}-\dfrac{1}{4}=\dfrac{1}{4}$ Here 4 is the LCM of 2 and 4.

We can see that the resulting denominator is not the product of 2 and 4, but it's the LCM of 2 and 4.

So, it's prooved that it's not necessary that in adding or subtracting two or more fractions, resulting denominator is the product of the denominators of the terms.