Multiplication of odds vs. multiplication of probabilities

Suppose that the prior probability of a message being spam is $\dfrac{1}{7}$, and the conditional probability of a message triggering the spam filter is $k$ if it is spam but $\dfrac{k}{4}$ if it is not spam for some $k$.

Then, given the spam filter is triggered, the posterior probability the message is spam is $\dfrac{\frac{1}{7}\times k}{ \frac{1}{7}\times k+ \frac{6}{7}\times \frac{k}{4} } = \dfrac{4}{10}$.

This is equivalent to the same statement that if the prior odds for spam are $1:6$ and the likelihood ratio for the spam filter being triggered is $4:1$ then the posterior odds for spam given the spam filter is triggered is $4:6$.

Effectively you have said the earlier calculation is equivalent to $\dfrac{\dfrac{\frac{1}{7}\times k}{ \frac{1}{7}\times k+ \frac{6}{7}\times \frac{k}{4} }}{\dfrac{\frac{6}{7}\times \frac{k}{4}}{ \frac{1}{7}\times k+ \frac{6}{7}\times \frac{k}{4} }} = \dfrac {\dfrac{4}{10}}{\dfrac{6}{10}}$ or in a simplified form $\dfrac{1}{6} \times \dfrac{4}{1}=\dfrac{4}{6}$.