Smallest Total Stock of Fruits

Let total fruits = $T$.

• Mangoes $M = 0.4 T$
• Bananas $B$ and Apples $A$, so $A + B = 0.6 T$

Fruits sold:

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

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

Since $A = 0.6T - B$, substitute $A$: $3T - 4(0.6T - B) = 960 \implies 0.6T + 4B = 960 \implies 3T + 20B = 4800$

Since $3T = 4800 - 20B$, $3T$ must be divisible by 20, which implies $T$ is divisible by 20. Also, $A \ge 1 \implies 3T = 960 + 4A \ge 964 \implies T \ge 322$.

The smallest multiple of 20 greater than or equal to 322 is $T = 340$.

Check for $T = 340$: $A = \frac{3(340) - 960}{4} = 15 \ge 1$ $B = \frac{4800 - 3(340)}{20} = 189 \ge 1$ $0.4 A = 0.4(15) = 6$ (an integer).

Hence, the minimum possible stock size $T = 340$.