`25 mL` sample of distilled water, tap water and boiled water required, respectively, `1 mL,13 mL` and `5 mL` of soap solution to form permanent lather. The ratio of permanent to temprorary hardness in the tap water is

Options

A. `3:2`
B. `2:3`
C. `1:2`
D. `2:1`

Explanation

To solve the problem, we need to determine the ratio of permanent to temporary hardness in tap water based on the volumes of soap solution required for different water samples.

### Step-by-Step Solution:

1. **Identify the volumes of soap solution used:**
- Distilled water: 1 mL
- Tap water: 13 mL
- Boiled water: 5 mL

2. **Calculate the effective volume of soap solution used for tap water:**
- The total volume of soap solution used for tap water is 13 mL.
- Since 1 mL is used for distilled water, the effective volume used for tap water is:
\[ \text{Effective volume for tap water} = 13 \, \text{mL} - 1 \, \text{mL} = 12 \, \text{mL} \]

3. **Determine the permanent hardness from boiled water:**
- The total volume of soap solution used for boiled water is 5 mL.
- Again, since 1 mL is used for distilled water, the effective volume used for boiled water is:
\[ \text{Effective volume for boiled water} = 5 \, \text{mL} - 1 \, \text{mL} = 4 \, \text{mL} \]
- This 4 mL represents the permanent hardness.

4. **Set up the equation for total hardness:**
- Let \( T \) be the temporary hardness and \( P \) be the permanent hardness.
- From the previous calculations, we have:
\[ T + P = 12 \, \text{mL} \quad \text{(from tap water)} \]
- We also know that:
\[ P = 4 \, \text{mL} \quad \text{(from boiled water)} \]

5. **Substitute the value of permanent hardness into the total hardness equation:**
- Substitute \( P = 4 \) mL into the equation:
\[ T + 4 = 12 \]
- Solving for \( T \):
\[ T = 12 - 4 = 8 \, \text{mL} \]

6. **Calculate the ratio of temporary to permanent hardness:**
- The ratio of temporary hardness \( T \) to permanent hardness \( P \) is:
\[ \text{Ratio} = \frac{T}{P} = \frac{8}{4} = 2 \]
- Therefore, the ratio of permanent to temporary hardness is:
\[ \text{Ratio of Permanent to Temporary} = \frac{P}{T} = \frac{4}{8} = \frac{1}{2} \]

### Final Answer:
The ratio of permanent to temporary hardness in the tap water is \( 1:2 \).

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