How does addition in the denominator work?
This is part of a series on common misconceptions.
Is this true or false?
\[ \dfrac{a}{b+c} = \dfrac{a}{b} + \dfrac{a}{c} \]
Why some people say it's true: It works when we add numerators like \(\dfrac b a + \dfrac c a = \dfrac{b+c} a \), so it's the same for denominators.
Why some people say it's false: Division is complicated, and you can't just add the things you're dividing by and still get a correct result.
The statement is \( \color{red}{\textbf{false}}\).
Proof:
Let us take the right hand side of the equation: \(\dfrac{a}{b} + \dfrac{a}{c}\).
Taking common denominator, we get\[\dfrac{a}{b} + \dfrac{a}{c} = \dfrac{ac + ab}{bc} =\dfrac{a(b + c)}{bc} =\frac{a}{b+c}\times\dfrac{(b + c)^2}{bc} \neq \frac{a}{b+c}.\]
The two terms are only equal in a very specific particular case.
Rebuttal: When \(a=0,\) \(\dfrac 0b+\dfrac 0c = \dfrac 0{b+c}.\)Reply: That just proves that it is true for this case only, which doesn't mean that it is true in general, i.e. we haven't proved it for all real values. In fact, let's test the hypothesis for \(a=20,\, b=1,\, c=4\) : we get \(\frac{20}{1+4}=\frac{20}5=4 \overset{?}{=} \frac{20}1 + \frac{20}4 = 20 + 5 = 25\), which are clearly not equal.
True or False:
If \(a, b, c,\text{ and } d \) are positive integers, then
\[ \frac{a}{b} + \frac{c}{d} = \frac{a+c}{b+d}.\]
See Also