If 2A = 3B = 4C then `A : B : C` = ?

Options

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

Answer

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

### Step-by-Step Solution:

1. **Set a Common Variable**:
Since \(2A = 3B = 4C\), we can set all of them equal to a common variable, say \(k\).
\[ 2A = k, \quad 3B = k, \quad 4C = k \]

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 the Ratio A : B : C**:
Now, we can write the ratio \(A : B : C\) as:
\[ A : B : C = \frac{k}{2} : \frac{k}{3} : \frac{k}{4} \]

4. **Eliminate k**:
Since \(k\) is common in all three terms, we can eliminate it:
\[ A : B : C = \frac{1}{2} : \frac{1}{3} : \frac{1}{4} \]

5. **Find a Common Denominator**:
To simplify this ratio, we can find a common denominator. The least common multiple of 2, 3, and 4 is 12. Therefore, we can convert each fraction:
\[ A : B : C = \frac{1 \times 6}{12} : \frac{1 \times 4}{12} : \frac{1 \times 3}{12} = 6 : 4 : 3 \]

6. **Final Ratio**:
Thus, the final ratio \(A : B : C\) is:
\[ A : B : C = 6 : 4 : 3 \]