# 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 ?

A. 256

B. 432

C. 512

D. 640

E. None of these

**Answer: Option C **

__Solution(By Examveda Team)__

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

__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**

__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,

$$\eqalign{ & n : \frac{n}{2} = 16 : \frac{n}{8} \cr & \Rightarrow \frac{{{n^2}}}{8} = 8n \cr & \Rightarrow \frac{n}{8} = 8 \cr & \Rightarrow n = 64 \cr} $$

Therefore, total number of books distributed

= 8n

= 8 × 64

= 512

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