`a=2b=3c=4d` then `a:b:c:d` is

Options

A. `12:3:6:4`
B. `3:4:6:12`
C. `6:12:4:3`
D. `12:6:4:3`

Answer

To solve the problem where \( a = 2b = 3c = 4d \), we need to express \( a, b, c, \) and \( d \) in terms of a common variable and then find the ratio \( a:b:c:d \).

### Step-by-Step Solution:

1. **Set a Common Variable**:
Let's introduce a common variable \( k \) such that:
\[ a = 2k, \quad b = k, \quad c = \frac{2k}{3}, \quad d = \frac{2k}{4} \]
Here, we express \( b \), \( c \), and \( d \) in terms of \( k \).

2. **Express Each Variable**:
From the equations:
- Since \( a = 2b \), we can express \( b \) as:
\[ b = \frac{a}{2} \]
- From \( 2b = 3c \), we can express \( c \) as:
\[ c = \frac{2b}{3} \]
- From \( 3c = 4d \), we can express \( d \) as:
\[ d = \frac{3c}{4} \]

3. **Substituting Values**:
Now substituting \( b = k \) into the expressions for \( c \) and \( d \):
- For \( c \):
\[ c = \frac{2k}{3} \]
- For \( d \):
\[ d = \frac{3 \cdot \frac{2k}{3}}{4} = \frac{2k}{4} = \frac{k}{2} \]

4. **Finding the Ratios**:
Now we have:
\[ a = 2k, \quad b = k, \quad c = \frac{2k}{3}, \quad d = \frac{k}{2} \]
To find the ratio \( a:b:c:d \):
- We can express all variables with a common denominator. The least common multiple of the denominators (1, 1, 3, 2) is 6.
- Thus, we can rewrite:
\[ a = 2k \cdot 6 = 12k, \quad b = k \cdot 6 = 6k, \quad c = \frac{2k}{3} \cdot 6 = 4k, \quad d = \frac{k}{2} \cdot 6 = 3k \]

5. **Final Ratio**:
Therefore, the ratio \( a:b:c:d \) becomes:
\[ a:b:c:d = 12k : 6k : 4k : 3k \]
Simplifying this gives:
\[ 12 : 6 : 4 : 3 \]

### Conclusion:
The final answer is:
\[ \boxed{12 : 6 : 4 : 3} \]