`25mL` samples of distiled water, tap water and boiled water required, respectively, `1mL, 13mL` and `5mL` of soap solution to form a permanent lather. The ratio of temporary to permanent 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 temporary to permanent hardness in tap water based on the amount of soap solution required to form a permanent lather. Here's a step-by-step breakdown of the solution:

### Step 1: Understand the Soap Solution Usage
We have three types of water: distilled water, tap water, and boiled water. The amount of soap solution used to form a permanent lather is given as follows:
- Distilled water: 1 mL
- Tap water: 13 mL
- Boiled water: 5 mL

### Step 2: Calculate Effective Soap Usage
The effective soap usage is the amount of soap that is used to overcome the hardness of the water. The effective soap usage can be calculated as follows:
- For distilled water: Since it has no hardness, the effective soap usage is 0 mL.
- For tap water: Effective soap usage = Total soap used - Soap used for permanent hardness
- For boiled water: Effective soap usage = Total soap used - Soap used for permanent hardness

### Step 3: Determine the Permanent Hardness
From the data, we can summarize:
- Distilled water: 0 mL (no hardness)
- Tap water: 13 mL of soap used, so effective soap = 13 mL - Permanent hardness
- Boiled water: 5 mL of soap used, so effective soap = 5 mL - Permanent hardness

Now, we know:
- The effective soap used for boiled water is 4 mL (5 mL - 1 mL).
- Therefore, the permanent hardness in boiled water is 4 mL.

### Step 4: Calculate Temporary Hardness
For tap water:
- The effective soap usage for tap water is 12 mL (13 mL - 1 mL).
- Permanent hardness is 4 mL (from boiled water).
- Therefore, temporary hardness can be calculated as:
\[ \text{Temporary Hardness} = \text{Effective Soap Usage} - \text{Permanent Hardness} \]
\[ \text{Temporary Hardness} = 12 \text{ mL} - 4 \text{ mL} = 8 \text{ mL} \]

### Step 5: Calculate the Ratio of Temporary to Permanent Hardness
Now that we have both temporary and permanent hardness values for tap water:
- Temporary Hardness = 8 mL
- Permanent Hardness = 4 mL

The ratio of temporary to permanent hardness is:
\[ \text{Ratio} = \frac{\text{Temporary Hardness}}{\text{Permanent Hardness}} = \frac{8 \text{ mL}}{4 \text{ mL}} = 2:1 \]

### Conclusion
The ratio of temporary to permanent hardness in tap water is **2:1**.

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