If x : y : z = 3 : 4 : 9, then what will be the value of (2x + 3y + z) : (2z + x – 3y)?

Options

A. `3 : 1`
B. `2 : 1`
C. `5 : 3`
D. `7 : 2`

Answer

To solve the problem, we need to find the value of the ratio \( (2x + 3y + z) : (2z + x - 3y) \) given that \( x : y : z = 3 : 4 : 9 \).

### Step-by-step Solution:

1. **Express x, y, and z in terms of a common variable:**
Since \( x : y : z = 3 : 4 : 9 \), we can express:
- \( x = 3k \)
- \( y = 4k \)
- \( z = 9k \)
where \( k \) is a positive constant.

2. **Substitute x, y, and z into the expressions:**
We need to evaluate:
- \( 2x + 3y + z \)
- \( 2z + x - 3y \)

Substituting the values:
- \( 2x = 2(3k) = 6k \)
- \( 3y = 3(4k) = 12k \)
- \( z = 9k \)

So,
\[ 2x + 3y + z = 6k + 12k + 9k = 27k \]

Now for the second expression:
- \( 2z = 2(9k) = 18k \)
- \( x = 3k \)
- \( 3y = 3(4k) = 12k \)

Thus,
\[ 2z + x - 3y = 18k + 3k - 12k = 9k \]

3. **Form the ratio:**
Now we can form the ratio:
\[ \frac{2x + 3y + z}{2z + x - 3y} = \frac{27k}{9k} \]

Simplifying this gives:
\[ \frac{27k}{9k} = \frac{27}{9} = 3 \]

4. **Express the ratio in standard form:**
The ratio can be expressed as:
\[ 3 : 1 \]

### Final Answer:
Thus, the value of \( (2x + 3y + z) : (2z + x - 3y) \) is \( 3 : 1 \).