A fruit seller has a stock of mangoes, bananas and apples with at least one fruit of each type. At the beginning of a day, the number of mangoes make up 40% of his stock. That day, he sells half of the mangoes, 96 bananas and 40% of the apples. At the end of the day, he ends up selling 50% of the fruits. The smallest possible total number of fruits in the stock at the beginning of the day is

Question

Accepted Answer

### Step 1: Define the variables  
Let the total initial stock of fruits be $$ s $$.  
- Number of mangoes = $$ 40\% $$ of $$ s = \frac{2s}{5} $$.  
- Number of bananas = $$ b $$.  
- Number of apples = $$ a $$.

Thus, the total stock at the beginning is:  
$$
s = \frac{2s}{5} + b + a.
$$

### Step 2: Analyze the fruits sold  
The fruit seller sells:  
1. Half of the mangoes: $$ \frac{1}{2} 	imes \frac{2s}{5} = \frac{s}{5} $$.  
2. 96 bananas.  
3. 40% of the apples: $$ 0.4a = \frac{4a}{10} = \frac{2a}{5} $$.

The total number of fruits sold is given as $$ 50\% $$ of the initial stock $$ s $$:  
$$
	ext{Fruits sold} = \frac{s}{5} + 96 + \frac{2a}{5} = \frac{s}{2}.
$$

### Step 3: Solve for $$ s $$  
Rearranging the equation:  
$$
\frac{s}{5} + 96 + \frac{2a}{5} = \frac{s}{2}.
$$  
Multiply through by 10 to eliminate fractions:  
$$
2s + 960 + 4a = 5s.
$$  
Simplify:  
$$
3s = 4a + 960.
$$  
Thus,  
$$
s = \frac{4a + 960}{3}.
$$

### Step 4: Find the smallest possible value of $$ s $$  
For $$ s $$ to be an integer, $$ (4a + 960) $$ must be divisible by 3. Hence, $$ a $$ must satisfy this condition. Additionally, since $$ a $$ is multiplied by $$ 0.4 = \frac{2}{5} $$, it must also be a multiple of 5 for $$ \frac{2a}{5} $$ to remain an integer.

Thus, $$ a $$ must be a common multiple of both 3 and 5 (i.e., a multiple of their LCM, which is 15). The smallest value of $$ a $$ is **15**.

### Step 5: Calculate $$ s $$ for $$ a = 15 $$  
Substitute $$ a = 15 $$:  
$$
s = \frac{4(15) + 960}{3}.
$$  
Simplify:  
$$
s = \frac{60 + 960}{3} = \frac{1020}{3} = 340.
$$

### Final Answer:  
The smallest possible total number of fruits in the stock at the beginning of the day is **340**.