(a) If `A : B = 3 : 4` and `B : C = 6 : 7`, find :
(i) `A : B : C`
(ii) `A : C`
(b) If `A : B = 2 : 5` and `A : C = 3 : 4`, find : `A : B : C`.

Answer

Let's solve the problem step by step.

### Part (a)

Given:
1. \( A : B = 3 : 4 \)
2. \( B : C = 6 : 7 \)

#### Step 1: Equalize the ratios
To find \( A : B : C \), we need to express both ratios in terms of a common value for \( B \).

The ratio \( A : B \) gives \( B = 4k \) for some constant \( k \).
The ratio \( B : C \) gives \( B = 6m \) for some constant \( m \).

To find a common value for \( B \), we can find the least common multiple (LCM) of 4 and 6, which is 12.

#### Step 2: Calculate values of A, B, and C
Now, we can express \( A \), \( B \), and \( C \) in terms of this common value:

1. From \( A : B = 3 : 4 \):
- If \( B = 12 \), then \( A = \frac{3}{4} \times 12 = 9 \).

2. From \( B : C = 6 : 7 \):
- If \( B = 12 \), then \( C = \frac{7}{6} \times 12 = 14 \).

#### Step 3: Write the final ratio
Now we can write:
- \( A = 9 \)
- \( B = 12 \)
- \( C = 14 \)

Thus, the combined ratio \( A : B : C = 9 : 12 : 14 \).

#### Step 4: Find \( A : C \)
To find \( A : C \):
- \( A : C = 9 : 14 \).

### Part (b)

Given:
1. \( A : B = 2 : 5 \)
2. \( A : C = 3 : 4 \)

#### Step 1: Equalize the ratios
Similar to part (a), we need to express both ratios in terms of a common value for \( A \).

From \( A : B = 2 : 5 \), we can express \( A = 2n \) and \( B = 5n \) for some constant \( n \).

From \( A : C = 3 : 4 \), we can express \( A = 3p \) and \( C = 4p \) for some constant \( p \).

To find a common value for \( A \), we can find the least common multiple (LCM) of 2 and 3, which is 6.

#### Step 2: Calculate values of A, B, and C
Now, we can express \( A \), \( B \), and \( C \) in terms of this common value:

1. From \( A : B = 2 : 5 \):
- If \( A = 6 \), then \( B = \frac{5}{2} \times 6 = 15 \).

2. From \( A : C = 3 : 4 \):
- If \( A = 6 \), then \( C = \frac{4}{3} \times 6 = 8 \).

#### Step 3: Write the final ratio
Now we can write:
- \( A = 6 \)
- \( B = 15 \)
- \( C = 8 \)

Thus, the combined ratio \( A : B : C = 6 : 15 : 8 \).

### Summary of Answers
- (i) \( A : B : C = 9 : 12 : 14 \)
- (ii) \( A : C = 9 : 14 \)
- (b) \( A : B : C = 6 : 15 : 8 \)