If 2A = 3 B = 8 C , What is A : B : C ?

Options

A. `8 : 3 : 2 `
B. ` 8 : 4 : 3`
C. ` 2 : 3 : 8`
D. ` 12 : 8 : 3`

Answer

To solve the problem where \(2A = 3B = 8C\), we need to express the ratios of \(A\), \(B\), and \(C\) in a simplified form.

### Step-by-Step Solution:

1. **Set the common value**:
Let \(k\) be the common value such that:
\[ 2A = k, \quad 3B = k, \quad 8C = k \]

2. **Express \(A\), \(B\), and \(C\) in terms of \(k\)**:
From the equations above, we can express \(A\), \(B\), and \(C\) as:
\[ A = \frac{k}{2}, \quad B = \frac{k}{3}, \quad C = \frac{k}{8} \]

3. **Find a common denominator**:
To find the ratio \(A : B : C\), we need to express all terms with a common denominator. The least common multiple (LCM) of the denominators \(2\), \(3\), and \(8\) is \(24\).

4. **Convert each term to have the common denominator**:
- For \(A\):
\[ A = \frac{k}{2} = \frac{12k}{24} \]
- For \(B\):
\[ B = \frac{k}{3} = \frac{8k}{24} \]
- For \(C\):
\[ C = \frac{k}{8} = \frac{3k}{24} \]

5. **Write the ratio**:
Now we can write the ratio \(A : B : C\) as:
\[ A : B : C = 12k : 8k : 3k \]

6. **Simplify the ratio**:
Since \(k\) is common in all terms, we can cancel it out:
\[ A : B : C = 12 : 8 : 3 \]

7. **Final simplification**:
To simplify further, we can divide each term by the greatest common divisor (GCD), which is \(1\) in this case, so the ratio remains:
\[ A : B : C = 12 : 8 : 3 \]

### Final Answer:
The ratio \(A : B : C\) is \(12 : 8 : 3\).