In what ratio should a 15% solution of acetic acid be mixed with a 3% solution of the acid to prepare a 10% solution (all percentage are mass/mass percentage) :

Options

A. `7:3`
B. `5:7`
C. `7:5`
D. `7:10`

Explanation

To solve the problem of mixing a 15% solution of acetic acid with a 3% solution to obtain a 10% solution, we can follow these steps:

### Step 1: Define Variables
Let:
- \( V_1 \) = volume of the 15% acetic acid solution
- \( V_2 \) = volume of the 3% acetic acid solution
- \( V_3 \) = total volume of the resulting 10% acetic acid solution

### Step 2: Set Up the Equation for Mass of Acetic Acid
The mass of acetic acid in each solution can be expressed as follows:
- Mass of acetic acid from the 15% solution = \( \frac{15}{100} \times V_1 \)
- Mass of acetic acid from the 3% solution = \( \frac{3}{100} \times V_2 \)

The total mass of acetic acid in the final solution (10% solution) is given by:
- Mass of acetic acid in the 10% solution = \( \frac{10}{100} \times V_3 \)

### Step 3: Relate the Volumes
Since \( V_3 \) is the total volume of the mixture, we have:
\[ V_3 = V_1 + V_2 \]

### Step 4: Substitute \( V_3 \) in the Mass Equation
Substituting \( V_3 \) into the mass equation gives:
\[ \frac{15}{100} V_1 + \frac{3}{100} V_2 = \frac{10}{100} (V_1 + V_2) \]

### Step 5: Simplify the Equation
Multiply through by 100 to eliminate the fractions:
\[ 15V_1 + 3V_2 = 10(V_1 + V_2) \]

Expanding the right side:
\[ 15V_1 + 3V_2 = 10V_1 + 10V_2 \]

### Step 6: Rearrange the Equation
Rearranging the terms gives:
\[ 15V_1 - 10V_1 = 10V_2 - 3V_2 \]
\[ 5V_1 = 7V_2 \]

### Step 7: Find the Ratio
Dividing both sides by \( V_2 \) and rearranging gives:
\[ \frac{V_1}{V_2} = \frac{7}{5} \]

### Conclusion
Thus, the ratio of the volumes of the 15% solution to the 3% solution is:
\[ \text{Ratio } V_1 : V_2 = 7 : 5 \]