If 2A = 3B = 4C, the find A : B C .

Options

A. `2 : 3: 4`
B. `4 : 3 : 2`
C. `6 : 4 : 3 `
D. `3 :4:6`

Answer

To solve the problem where \( 2A = 3B = 4C \) and we need to find the ratio \( A : B : C \), we can follow these steps:

### Step 1: Set a common variable
Since \( 2A = 3B = 4C \), we can set them equal to a common variable, say \( k \). Thus, we have:
\[ 2A = k, \quad 3B = k, \quad 4C = k \]

### Step 2: Express A, B, and C in terms of k
Now, we can express \( A \), \( B \), and \( C \) in terms of \( k \):
\[ A = \frac{k}{2}, \quad B = \frac{k}{3}, \quad C = \frac{k}{4} \]

### Step 3: Find a common denominator
To find the ratio \( A : B : C \), we need to express \( A \), \( B \), and \( C \) with a common denominator. The least common multiple of 2, 3, and 4 is 12.

### Step 4: Rewrite A, B, and C with the common denominator
Now we can rewrite \( A \), \( B \), and \( C \) as follows:
\[ A = \frac{k}{2} = \frac{6k}{12}, \quad B = \frac{k}{3} = \frac{4k}{12}, \quad C = \frac{k}{4} = \frac{3k}{12} \]

### Step 5: Write the ratio
Now we can write the ratio \( A : B : C \):
\[ A : B : C = 6k : 4k : 3k \]

### Step 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 required ratio \( A : B : C \) is:
\[ \boxed{6 : 4 : 3} \]