Free books were distributed equally among children of a class. The number of books each child got was one-eighth of the number of children. Had the number of children been half, each child would have got $16$ books. How many books were distributed in total?

Let the total number of children in the class be $N$.
According to the first condition, the number of books each child got is $\frac{N}{8}$.

$$\text{Total number of books}=N\times \frac{N}{8}=\frac{N^2}{8}$$

According to the second condition, if the number of children was half $\left(\frac{N}{2}\right)$, each child would get $16$ books:

$$\text{Total number of books}=\frac{N}{2}\times 16=8N$$

Since the total number of books remains constant, we equate the two expressions:

$$\frac{N^2}{8}=8N$$

Since $N\ne 0$, we can divide by $N$:

$$\frac{N}{8}=8⟹N=64$$

The total number of books distributed is:

$$\text{Total Books}=8\times 64=512$$