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In the Corollary at pag 407 of Young persons guide to canonical singularities there is a formula to compute the contributions $c_q(D)$ to Riemann-Roch of a divisor $D$ passing through a point $q\in X$, where $X$ is a normal surface, $q\in X$ is a quotient singularity of type $\frac{1}{r}(a_1,a_2)$, and $L = \mathcal{O}_X(D)$ is of type $i(\frac{1}{r}(a_1,a_2))$ at $q$ for $i = 0,...,r-1$.

Now, let us take the simplest case $r=2, a_1 = a_2 = 1$, $i = 1$. By the formula in the Corollary at pag 407 we get $c_q(D) = \sigma_1 - \sigma_0 = -\frac{1}{4}-\frac{1}{4} = -\frac{1}{2}$.

On the other hand by the formula in the end of pag 409 we get $c_q(D)=-\frac{1}{4}$.

Does anyone know what is wrong here? Am I misunderstanding anything?

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The last formula on page 409 clearly has a typo. There should be no $r$ in the denominator. If you look at the computation, (or even better, do it yourself!!), then you see that $r$ never appears in the denominator (they are always explicit numbers). In particular, this formula is based on the formula in the Proposition in (8.9) where it appears correctly as $-\frac{i(r-i)}2$. In fact, it appears correctly in the proof of the theorem with which you're having trouble (see the two lines starting with (II) just before (9.4) on page 410).

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