Stock Optimization and Percentage Sales

Let total initial stock = $T$. $M = 0.4 T$, $B$, and $A$ are the number of mangoes, bananas, and apples. $M + B + A = T \implies B + A = 0.6 T$.

Fruits sold:

• Mangoes sold = $0.5 M = 0.2 T$
• Bananas sold = 96
• Apples sold = $0.4 A$

Total sold = $0.5 T$: $0.2 T + 96 + 0.4 A = 0.5 T \implies 0.3 T - 0.4 A = 96$ $3T - 4A = 960 \implies T = 320 + \frac{4}{3}A$

For $T$ to be an integer, $A$ must be a multiple of 3. Since $40\%$ of apples are sold, $0.4A = \frac{2}{5}A$ must be an integer, so $A$ must also be a multiple of 5. Thus, $A$ must be a multiple of 15.

Also, initial bananas $B = 0.6T - A \ge 96$ (since 96 bananas are sold): $0.6\left(320 + \frac{4}{3}A\right) - A \ge 96 \implies 192 - 0.2 A \ge 96 \implies A \le 480$

To minimize $T = 320 + \frac{4}{3}A$, we minimize $A$. Smallest positive multiple of 15 for $A$ is $A = 15$.

Smallest possible $T = 320 + \frac{4}{3}(15) = 340$.