A vendor has two containers containing milk and water solutions of volume 10 liters and 20 liters. What would be the minimum concetration (%) of milk in any of the containers so that he could mix them to get an 80% milk solution ?

Answer

To solve the problem, we need to find the minimum concentration of milk in either of the two containers so that when mixed, the resulting solution has an 80% concentration of milk. Let's break down the steps:

### Step 1: Define Variables
Let:
- \( x \) = concentration of milk in the first container (10 liters)
- \( y \) = concentration of milk in the second container (20 liters)

### Step 2: Express the Amount of Milk
The amount of milk in each container can be expressed as:
- Amount of milk in the first container = \( \frac{x}{100} \times 10 \) liters
- Amount of milk in the second container = \( \frac{y}{100} \times 20 \) liters

### Step 3: Total Amount of Milk
When both containers are mixed, the total amount of milk will be:
\[ \text{Total milk} = \frac{x}{100} \times 10 + \frac{y}{100} \times 20 \]

### Step 4: Total Volume of the Mixture
The total volume of the mixture is:
\[ \text{Total volume} = 10 + 20 = 30 \text{ liters} \]

### Step 5: Set Up the Equation for 80% Concentration
We want the resulting mixture to have an 80% concentration of milk:
\[ \frac{\frac{x}{100} \times 10 + \frac{y}{100} \times 20}{30} = 0.8 \]

### Step 6: Simplify the Equation
Multiply both sides by 30:
\[ \frac{x}{100} \times 10 + \frac{y}{100} \times 20 = 24 \]
This simplifies to:
\[ \frac{x}{10} + \frac{2y}{10} = 24 \]
Multiplying through by 10 gives:
\[ x + 2y = 240 \]

### Step 7: Analyze the Equation
We need to minimize \( x \) or \( y \). To find the minimum concentration in either container, we can express \( x \) in terms of \( y \):
\[ x = 240 - 2y \]

### Step 8: Determine Maximum Values for \( y \)
The maximum possible concentration for \( y \) (the second container) is 100% (pure milk):
\[ y = 100 \Rightarrow x = 240 - 2(100) = 40 \]

### Step 9: Determine Maximum Values for \( x \)
The maximum possible concentration for \( x \) (the first container) is also 100%:
\[ x = 100 \Rightarrow 100 + 2y = 240 \Rightarrow 2y = 140 \Rightarrow y = 70 \]

### Step 10: Compare Values
Now we have two concentrations:
- For \( y = 100 \), \( x = 40 \)
- For \( x = 100 \), \( y = 70 \)

The minimum concentration of milk in either container is:
\[ \text{Minimum concentration} = \min(40, 70) = 40\% \]

### Final Answer
The minimum concentration of milk in any of the containers so that the vendor can mix them to get an 80% milk solution is **40%**.

---