Free notebooks were distributed equally among n children. Each child received (n/8) notebooks. If the number of children had been half, each would have received 16 notebooks. How many notebooks were distributed in total?

Given data

• Number of children = n; each got n/8 notebooks.
• Total notebooks T = n × (n/8) = n^2/8.
• If children were n/2, each would get 16 ⇒ T ÷ (n/2) = 16.

Concept / Approach

• Equate the two expressions for the same total T to solve for n.

Step-by-step calculationT/(n/2) = 16 ⇒ 2T/n = 16 ⇒ T = 8nBut also T = n^2/8 ⇒ n^2/8 = 8nn^2 = 64n ⇒ n = 64 (n > 0)T = 8n = 8 × 64 = 512

VerificationOriginal per-child = n/8 = 64/8 = 8; 64 children × 8 = 512 ✔With n/2 = 32 children: 512/32 = 16 each ✔

Final Answer512