A chemist has 20% and 30% concentration of nitric acid with him. In an experiment, he mixed x mI of 20% concentration and 9 mI of 30% concentration in one container and in another container he mixed 4 mI of the 20% concentration with x ml of the 30% conentration. Surprisingly, the concentration of acid in both containers came out to be same. What is the value of X ( in ml) ?

Answer

To solve the problem step by step, we will first define the variables and then set up the equations based on the concentrations and volumes given.

### Step 1: Define the Variables
Let:
- \( x \) = volume of 20% nitric acid in the first container (in mL)
- 9 mL = volume of 30% nitric acid in the first container
- 4 mL = volume of 20% nitric acid in the second container
- \( x \) mL = volume of 30% nitric acid in the second container

### Step 2: Calculate the Amount of Solute in Each Container
The amount of solute (nitric acid) in each solution can be calculated using the formula:
\[ \text{Amount of solute} = \text{Concentration} \times \text{Volume} \]

**For the first container:**
- Amount of solute from 20% solution:
\[ S_1 = 0.20 \times x = 0.2x \text{ mL} \]
- Amount of solute from 30% solution:
\[ S_2 = 0.30 \times 9 = 2.7 \text{ mL} \]
- Total solute in the first container:
\[ \text{Total solute in A} = S_1 + S_2 = 0.2x + 2.7 \text{ mL} \]

**For the second container:**
- Amount of solute from 20% solution:
\[ S_3 = 0.20 \times 4 = 0.8 \text{ mL} \]
- Amount of solute from 30% solution:
\[ S_4 = 0.30 \times x = 0.3x \text{ mL} \]
- Total solute in the second container:
\[ \text{Total solute in B} = S_3 + S_4 = 0.8 + 0.3x \text{ mL} \]

### Step 3: Set Up the Equation for Concentrations
Since the concentrations of both containers are the same, we can set up the equation:
\[ \frac{0.2x + 2.7}{x + 9} = \frac{0.8 + 0.3x}{4 + x} \]

### Step 4: Cross Multiply to Solve for \( x \)
Cross-multiplying gives:
\[ (0.2x + 2.7)(4 + x) = (0.8 + 0.3x)(x + 9) \]

### Step 5: Expand Both Sides
Expanding both sides:
- Left side:
\[ 0.2x \cdot 4 + 0.2x \cdot x + 2.7 \cdot 4 + 2.7 \cdot x = 0.8x + 2.7x + 10.8 \]
\[ 0.8x + 0.2x^2 + 10.8 + 2.7x = 0.2x^2 + 3.5x + 10.8 \]

- Right side:
\[ 0.8 \cdot x + 0.8 \cdot 9 + 0.3x \cdot x + 0.3x \cdot 9 = 0.3x^2 + 0.8x + 2.7x + 7.2 \]
\[ 0.3x^2 + 3.5x + 7.2 \]

### Step 6: Set Up the Final Equation
Setting both sides equal:
\[ 0.2x^2 + 3.5x + 10.8 = 0.3x^2 + 3.5x + 7.2 \]

### Step 7: Rearranging the Equation
Rearranging gives:
\[ 0.2x^2 - 0.3x^2 + 10.8 - 7.2 = 0 \]
\[ -0.1x^2 + 3.6 = 0 \]
\[ 0.1x^2 = 3.6 \]
\[ x^2 = 36 \]
\[ x = 6 \text{ mL} \]

### Final Answer
The value of \( x \) is \( 6 \) mL.

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