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 ?

Explanation:-

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

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Explaination 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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Explaination 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,
$$\eqalign{ & n : \frac{n}{2} = 16 : \frac{n}{8} \cr & \Rightarrow \frac{{{n^2}}}{8} = 8n \cr & \Rightarrow \frac{n}{8} = 8 \cr & \Rightarrow n = 8 \cr}$$
Therefore, total number of books distributed
=8n = 8×64 = 512