If 2 A = 3B = 4C, find `A : B : C`

Answer

To solve the problem of finding the ratio \( A : B : C \) given that \( 2A = 3B = 4C \), we can follow these steps:

### Step-by-Step Solution:

1. **Set a Common Variable**:
Let's denote the common value of \( 2A \), \( 3B \), and \( 4C \) as \( k \). Therefore, we can write:
\[ 2A = k, \quad 3B = k, \quad 4C = k \]

2. **Express A, B, and C in terms of k**:
From the equations above, we can express \( A \), \( B \), and \( C \) in terms of \( k \):
\[ A = \frac{k}{2}, \quad B = \frac{k}{3}, \quad C = \frac{k}{4} \]

3. **Find a Common Denominator**:
To find the ratio \( A : B : C \), we need to express all three variables with a common denominator. The least common multiple (LCM) of the denominators \( 2, 3, \) and \( 4 \) is \( 12 \).

4. **Convert A, B, and C**:
Now, we convert \( A \), \( B \), and \( C \) to have a denominator of \( 12 \):
\[ A = \frac{k}{2} = \frac{6k}{12}, \quad B = \frac{k}{3} = \frac{4k}{12}, \quad C = \frac{k}{4} = \frac{3k}{12} \]

5. **Write the Ratio**:
Now we can write the ratio:
\[ A : B : C = 6k : 4k : 3k \]

6. **Simplify the Ratio**:
Since \( k \) is common in all terms, we can simplify the ratio:
\[ A : B : C = 6 : 4 : 3 \]

### Final Answer:
Thus, the ratio \( A : B : C \) is:
\[ \boxed{6 : 4 : 3} \]