The weight of an iron bucket increases by `33.33%` when filled with water to `50%` of its capacity . Which of these may be `50%` of the weight of the bucket when it is completely filled with water (assume the weight of bucket and its capacity in Kg. to be integers ) ?

Options

A. `136.5` kg
B. `135.5` kg
C. `122.5` kg
D. `117` kg

Answer

To solve the problem, we need to understand the relationship between the weight of the bucket, the water it holds, and the percentage increase in weight when filled to 50% of its capacity.

### Step-by-Step Solution:

1. **Understanding the Percentage Increase**:
The problem states that the weight of the bucket increases by 33.33% when filled with water to 50% of its capacity. This means that the weight of the water added is equal to 33.33% of the weight of the bucket.

2. **Let the Weight of the Bucket be W**:
Let the weight of the empty bucket be \( W \) kg.

3. **Calculating the Weight of Water**:
When the bucket is filled to 50% of its capacity, the weight of the water added is \( 0.5C \), where \( C \) is the total capacity of the bucket in kg. According to the problem, this weight of water causes the total weight to increase by 33.33% of \( W \):
\[ 0.5C = \frac{1}{3}W \]

4. **Finding the Relationship Between W and C**:
Rearranging the equation gives:
\[ C = \frac{2}{3}W \]
This means that the total capacity of the bucket is two-thirds of the weight of the bucket.

5. **Finding 50% of the Weight of the Bucket When Filled with Water**:
When the bucket is completely filled with water, the weight of the water will be \( C \). Therefore, the total weight of the bucket when filled with water is:
\[ W + C = W + \frac{2}{3}W = \frac{5}{3}W \]
Now, we need to find 50% of this total weight:
\[ \text{50% of total weight} = \frac{1}{2} \times \frac{5}{3}W = \frac{5}{6}W \]

6. **Finding Possible Values for W**:
Since \( W \) must be an integer, \( \frac{5}{6}W \) must also be an integer. This means \( W \) must be a multiple of 6. Possible integer values for \( W \) could be 6, 12, 18, 24, 30, etc.

7. **Calculating 50% of the Weight for Possible W Values**:
- If \( W = 6 \), then \( \frac{5}{6}W = 5 \)
- If \( W = 12 \), then \( \frac{5}{6}W = 10 \)
- If \( W = 18 \), then \( \frac{5}{6}W = 15 \)
- If \( W = 24 \), then \( \frac{5}{6}W = 20 \)
- If \( W = 30 \), then \( \frac{5}{6}W = 25 \)
- If \( W = 36 \), then \( \frac{5}{6}W = 30 \)
- If \( W = 42 \), then \( \frac{5}{6}W = 35 \)
- If \( W = 48 \), then \( \frac{5}{6}W = 40 \)
- If \( W = 54 \), then \( \frac{5}{6}W = 45 \)
- If \( W = 60 \), then \( \frac{5}{6}W = 50 \)

### Conclusion:
The possible values for 50% of the weight of the bucket when it is completely filled with water are 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50 kg.