A sample of clay contains `50%` silica and `10%` water. The sample is partially dried by which it loses 8 gm of water. If the percentage of silica in the partially dried clay is 52, what is the percentage of water is the partially dried clay?

Options

A. `2.0%`
B. `6.4%`
C. `10.4%`
D. `2.4%`

Explanation

To solve the problem step by step, we will follow the information given about the clay sample and the changes that occur when it is partially dried.

### Step 1: Determine the initial composition of the clay sample.
The clay sample contains:
- 50% silica
- 10% water
- Therefore, the remaining components make up 40% (which we will denote as "other components").

Let the total weight of the clay sample be \( x \) grams.

### Step 2: Calculate the weight of silica and water in the original sample.
- Weight of silica = \( 50\% \) of \( x \) = \( 0.5x \) grams
- Weight of water = \( 10\% \) of \( x \) = \( 0.1x \) grams

### Step 3: Account for the loss of water.
The sample loses 8 grams of water. Therefore, the weight of water after drying becomes:
- Weight of water after drying = \( 0.1x - 8 \) grams

### Step 4: Set up the equation for the weight of silica in the partially dried clay.
After drying, the total weight of the clay becomes:
- Total weight of partially dried clay = \( x - 8 \) grams

According to the problem, the percentage of silica in the partially dried clay is 52%. Therefore, the weight of silica in the partially dried clay can be expressed as:
- Weight of silica in partially dried clay = \( 52\% \) of \( (x - 8) \) = \( 0.52(x - 8) \) grams

### Step 5: Set the weights of silica equal to each other.
Since the weight of silica does not change during the drying process, we can set up the equation:
\[ 0.5x = 0.52(x - 8) \]

### Step 6: Solve the equation for \( x \).
Expanding the right side:
\[ 0.5x = 0.52x - 4.16 \]
Rearranging gives:
\[ 0.5x - 0.52x = -4.16 \]
\[ -0.02x = -4.16 \]
\[ x = \frac{4.16}{0.02} = 208 \text{ grams} \]

### Step 7: Calculate the initial weight of water.
Now that we have \( x \), we can find the initial weight of water:
\[ \text{Weight of water} = 0.1x = 0.1 \times 208 = 20.8 \text{ grams} \]

### Step 8: Calculate the weight of water after drying.
After losing 8 grams of water:
\[ \text{Weight of water after drying} = 20.8 - 8 = 12.8 \text{ grams} \]

### Step 9: Calculate the total weight of the partially dried clay.
The total weight of the partially dried clay is:
\[ \text{Total weight of partially dried clay} = 208 - 8 = 200 \text{ grams} \]

### Step 10: Calculate the percentage of water in the partially dried clay.
The percentage of water in the partially dried clay is given by:
\[ \text{Percentage of water} = \left(\frac{\text{Weight of water after drying}}{\text{Total weight of partially dried clay}}\right) \times 100 \]
\[ \text{Percentage of water} = \left(\frac{12.8}{200}\right) \times 100 = 6.4\% \]

### Final Answer:
The percentage of water in the partially dried clay is **6.4%**.

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