0.1 and `5/8` of a bamboo are in mud and water respectively and the rest of length 2.75 m is above water. What is the length of the bamboo?

Options

A. 10 m
B. 30 m
C. 27.5 m
D. 20 m

Answer

To solve the problem, we need to find the total length of the bamboo given that certain fractions of it are submerged in mud and water, and the remaining length is above water.

### Step-by-Step Solution:

1. **Define the Total Length of the Bamboo**:
Let the total length of the bamboo be denoted as \( L \).

2. **Identify the Parts of the Bamboo**:
- The part of the bamboo in mud is \( 0.1L \).
- The part of the bamboo in water is \( \frac{5}{8}L \).
- The part of the bamboo above water is given as \( 2.75 \, \text{m} \).

3. **Set Up the Equation**:
The total length of the bamboo can be expressed as the sum of the parts in mud, in water, and above water:
\[ 0.1L + \frac{5}{8}L + 2.75 = L \]

4. **Combine the Terms**:
To combine the terms involving \( L \), we first convert \( 0.1 \) to a fraction:
\[ 0.1 = \frac{1}{10} \]
Now, we can express the equation as:
\[ \frac{1}{10}L + \frac{5}{8}L + 2.75 = L \]

5. **Find a Common Denominator**:
The common denominator for \( 10 \) and \( 8 \) is \( 40 \). We convert each term:
\[ \frac{1}{10}L = \frac{4}{40}L, \quad \frac{5}{8}L = \frac{25}{40}L \]
So the equation becomes:
\[ \frac{4}{40}L + \frac{25}{40}L + 2.75 = L \]

6. **Combine the Fractions**:
Combine the fractions on the left side:
\[ \frac{29}{40}L + 2.75 = L \]

7. **Isolate \( L \)**:
To isolate \( L \), subtract \( \frac{29}{40}L \) from both sides:
\[ 2.75 = L - \frac{29}{40}L \]
This simplifies to:
\[ 2.75 = \frac{11}{40}L \]

8. **Solve for \( L \)**:
Multiply both sides by \( \frac{40}{11} \) to solve for \( L \):
\[ L = 2.75 \times \frac{40}{11} \]

9. **Calculate the Value**:
Calculate \( L \):
\[ L = \frac{2.75 \times 40}{11} = \frac{110}{11} = 10 \, \text{m} \]

### Final Answer:
The total length of the bamboo is \( 10 \, \text{m} \).