A report states that for a distribution with classes 0-10, 10-20, 20-30, 30-40 the median class is 10-20 but the median is 24. Why is this impossible?

For the median class, $cf<\frac{n}{2}\le cf+f$, so the fraction $\frac{n/2-cf}{f}$ always lies between 0 and 1. Hence Median = $l+\left(\frac{n/2-cf}{f}\right)\times h$ lies between $l$ and $l+h$ — here between 10 and 20, never 24. The median need NOT equal the class mark (that happens only when $\frac{n}{2}$ sits exactly halfway through the class); the largest frequency defines the MODAL class, not the median class; and a grouped median is very often a decimal.