Free notebooks were distributed equally among children of a class. The number of notebooks each child got was one-eighth of the number of children. Had the number of children been half, each child would have got 16 notebooks. Total how many notebooks were distributed ?

Answer: Option C

Explanation 1
$$\begin{array}{rl}& \text{Let number of children}=n\\ & \text{Then, number of books each child will get }=\frac{n}{8}\\ & \text{Total books distributed}=n\times \frac{n}{8}=\frac{n^2}{8}\\ & \text{If the number children}=\frac{n}{2},\\ & \text{number of books each child will get}=16\\ & \text{Total books distributed }=\frac{n}{2}\times 16=8n\\ & \therefore \frac{n^2}{8}=8n\\ & \Rightarrow \frac{n}{8}=8\\ & \Rightarrow n=64\\ & \text{Total number of books distributed}\\ & =8n=8\times 64=512\end{array}$$

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Explanation 2
If number of children was half, each child would have got 16 books.
Therefore, actually each child got $$\frac{16}{2}$$ = 8 Books And the number of children is 8 × 8 = 64
Hence, total number of books distributed = 64 × 8 = 512

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Explanation 3
Let number of children $$=n$$
Then, number of books each child will get $$=\frac{n}{8}$$
If the number children $$=\frac{n}{2}$$,
number of books each child will get $$=16$$
More children, less books (indirect proportion). Therefore,
$$\begin{array}{rl}& n:\frac{n}{2}=16:\frac{n}{8}\\ & \Rightarrow \frac{n^2}{8}=8n\\ & \Rightarrow \frac{n}{8}=8\\ & \Rightarrow n=64\end{array}$$
Therefore, total number of books distributed
= 8n
= 8 × 64
= 512