If A : B: C = 2: 3 : 4 , then what is ` (A) /(B) : (B ) /( C ) : (C ) /(A) ` equal to?

Answer

To solve the problem, we need to find the ratio \( \frac{A}{B} : \frac{B}{C} : \frac{C}{A} \) given that \( A : B : C = 2 : 3 : 4 \).

### Step-by-Step Solution:

1. **Express A, B, and C in terms of a common variable:**
Since \( A : B : C = 2 : 3 : 4 \), we can express A, B, and C as:
\[ A = 2x, \quad B = 3x, \quad C = 4x \]
where \( x \) is a common multiplier.

**Hint:** Use a variable to represent the ratio parts for easier calculations.

2. **Calculate \( \frac{A}{B} \):**
\[ \frac{A}{B} = \frac{2x}{3x} = \frac{2}{3} \]

**Hint:** When dividing two fractions with the same variable, the variable cancels out.

3. **Calculate \( \frac{B}{C} \):**
\[ \frac{B}{C} = \frac{3x}{4x} = \frac{3}{4} \]

**Hint:** Again, the common variable cancels out, simplifying the fraction.

4. **Calculate \( \frac{C}{A} \):**
\[ \frac{C}{A} = \frac{4x}{2x} = \frac{4}{2} = 2 \]

**Hint:** Simplify the fraction by dividing the numerator and denominator by the common variable.

5. **Now, we have the ratios:**
\[ \frac{A}{B} : \frac{B}{C} : \frac{C}{A} = \frac{2}{3} : \frac{3}{4} : 2 \]

6. **Convert the ratios to a common denominator:**
The denominators are 3, 4, and 1. The least common multiple (LCM) of 3, 4, and 1 is 12.

- For \( \frac{2}{3} \):
\[ \frac{2}{3} \times \frac{4}{4} = \frac{8}{12} \]
- For \( \frac{3}{4} \):
\[ \frac{3}{4} \times \frac{3}{3} = \frac{9}{12} \]
- For \( 2 \):
\[ 2 \times \frac{12}{12} = \frac{24}{12} \]

**Hint:** To convert ratios to a common denominator, multiply each fraction by the appropriate factor.

7. **Combine the ratios:**
Now we have:
\[ \frac{A}{B} : \frac{B}{C} : \frac{C}{A} = 8 : 9 : 24 \]

**Final Answer:**
\[ \frac{A}{B} : \frac{B}{C} : \frac{C}{A} = 8 : 9 : 24 \]