When X is subtracted from the numbers `9,15 and 27`, The remainders are in continuted proportion. What is the value of X ?

Options

A. 8
B. 6
C. 4
D. None of these

Explanation

To solve the problem, we need to find the value of \( X \) such that when it is subtracted from the numbers \( 9, 15, \) and \( 27 \), the remainders are in continued proportion.

### Step-by-step Solution:

1. **Understanding Continued Proportion**:
When we say that the numbers \( a, b, c \) are in continued proportion, it means that:
\[ \frac{b}{a} = \frac{c}{b} \]
In our case, the numbers after subtracting \( X \) will be \( 9 - X, 15 - X, \) and \( 27 - X \).

2. **Setting up the Proportion**:
According to the definition of continued proportion, we can set up the equation:
\[ \frac{15 - X}{9 - X} = \frac{27 - X}{15 - X} \]

3. **Cross Multiplying**:
Cross multiplying gives us:
\[ (15 - X)(15 - X) = (9 - X)(27 - X) \]

4. **Expanding Both Sides**:
Expanding the left side:
\[ (15 - X)^2 = 225 - 30X + X^2 \]
Expanding the right side:
\[ (9 - X)(27 - X) = 243 - 36X + X^2 \]

5. **Setting the Equation**:
Now we set the two expansions equal to each other:
\[ 225 - 30X + X^2 = 243 - 36X + X^2 \]

6. **Simplifying the Equation**:
We can cancel \( X^2 \) from both sides:
\[ 225 - 30X = 243 - 36X \]
Rearranging gives:
\[ 36X - 30X = 243 - 225 \]
Which simplifies to:
\[ 6X = 18 \]

7. **Solving for \( X \)**:
Dividing both sides by 6:
\[ X = 3 \]

### Final Answer:
The value of \( X \) is \( 3 \).