If `A : B = 5 : 7` and `B : C = 6 : 11` then `A : B : C` = ?

Options

A. `30 : 42 : 55`
B. `30 : 42 : 77`
C. `35 : 49 : 66`
D. none of these

Answer

To solve the problem of finding the ratio \( A : B : C \) given \( A : B = 5 : 7 \) and \( B : C = 6 : 11 \), we can follow these steps:

### Step-by-Step Solution:

1. **Express the Ratios in Fraction Form**:
- From the ratio \( A : B = 5 : 7 \), we can express it as:
\[ \frac{A}{B} = \frac{5}{7} \]
- From the ratio \( B : C = 6 : 11 \), we can express it as:
\[ \frac{B}{C} = \frac{6}{11} \]

2. **Express A and C in terms of B**:
- From \( \frac{A}{B} = \frac{5}{7} \), we can rearrange to find \( A \):
\[ A = \frac{5}{7}B \]
- From \( \frac{B}{C} = \frac{6}{11} \), we can rearrange to find \( C \):
\[ C = \frac{11}{6}B \]

3. **Find a Common Value for B**:
- To combine the ratios, we need to express all terms in terms of a common value for \( B \). Let's assume \( B = 42 \) (the least common multiple of 7 and 6).

4. **Calculate A and C**:
- Substitute \( B = 42 \) into the equations for \( A \) and \( C \):
\[ A = \frac{5}{7} \times 42 = 30 \]
\[ C = \frac{11}{6} \times 42 = 77 \]

5. **Write the Final Ratio**:
- Now we have the values:
- \( A = 30 \)
- \( B = 42 \)
- \( C = 77 \)
- Therefore, the ratio \( A : B : C \) is:
\[ A : B : C = 30 : 42 : 77 \]

### Final Answer:
\[ A : B : C = 30 : 42 : 77 \]