(x-a):(x-b):(x-c)=11 :9 : 5, where `x=(a+b+c)/2`. What is the ratio of a, b, c ?

Options

A. 0.42230324074074
B. 0.12857638888889
C. 0.29733796296296
D. 0.25350694444444

Answer

To solve the problem, we need to find the ratio of \( a, b, c \) given that:

\[ (x-a):(x-b):(x-c) = 11:9:5 \]

and

\[ x = \frac{a+b+c}{2}. \]

### Step-by-Step Solution:

1. **Set Up the Ratios**:
We can express \( x-a \), \( x-b \), and \( x-c \) in terms of a common variable \( k \):
\[ x - a = 11k, \quad x - b = 9k, \quad x - c = 5k. \]

2. **Express \( a, b, c \)**:
Rearranging these equations gives:
\[ a = x - 11k, \quad b = x - 9k, \quad c = x - 5k. \]

3. **Substitute \( x \)**:
Since \( x = \frac{a+b+c}{2} \), we can substitute \( a, b, c \) into this equation:
\[ x = \frac{(x - 11k) + (x - 9k) + (x - 5k)}{2}. \]

4. **Combine the Terms**:
Simplifying the right-hand side:
\[ x = \frac{3x - 25k}{2}. \]

5. **Clear the Fraction**:
Multiply both sides by 2 to eliminate the fraction:
\[ 2x = 3x - 25k. \]

6. **Rearrange the Equation**:
Rearranging gives:
\[ 3x - 2x = 25k \implies x = 25k. \]

7. **Substitute \( x \) Back**:
Now substitute \( x = 25k \) back into the expressions for \( a, b, c \):
\[ a = 25k - 11k = 14k, \]
\[ b = 25k - 9k = 16k, \]
\[ c = 25k - 5k = 20k. \]

8. **Find the Ratio**:
The ratio of \( a, b, c \) is:
\[ a:b:c = 14k:16k:20k. \]

9. **Simplify the Ratio**:
Dividing each term by \( 2k \) gives:
\[ a:b:c = 7:8:10. \]

### Final Answer:
The ratio of \( a, b, c \) is \( 7:8:10 \).