If `A : B = 3 : 4` and `B : C = 6 : 5`, then `A : (A + C)` is equal to:

Options

A. `9:10`
B. `10:9`
C. `9:19`
D. `19:9`

Answer

To solve the problem, we need to find the ratio \( A : (A + C) \) given that \( A : B = 3 : 4 \) and \( B : C = 6 : 5 \).

### Step-by-Step Solution:

1. **Write the Ratios**:
We start with the given ratios:
\[ A : B = 3 : 4 \quad \text{(1)} \]
\[ B : C = 6 : 5 \quad \text{(2)} \]

2. **Express B in terms of A**:
From ratio (1), we can express \( A \) in terms of \( B \):
\[ A = \frac{3}{4}B \quad \text{(3)} \]

3. **Express C in terms of B**:
From ratio (2), we can express \( C \) in terms of \( B \):
\[ C = \frac{5}{6}B \quad \text{(4)} \]

4. **Substitute B in terms of A**:
To find \( A : (A + C) \), we need to express \( C \) in terms of \( A \). From equation (3), we can express \( B \) in terms of \( A \):
\[ B = \frac{4}{3}A \quad \text{(5)} \]
Now substitute equation (5) into equation (4):
\[ C = \frac{5}{6} \left(\frac{4}{3}A\right) = \frac{20}{18}A = \frac{10}{9}A \quad \text{(6)} \]

5. **Calculate A + C**:
Now we can find \( A + C \):
\[ A + C = A + \frac{10}{9}A = \frac{9}{9}A + \frac{10}{9}A = \frac{19}{9}A \quad \text{(7)} \]

6. **Find the Ratio A : (A + C)**:
Now we can find the ratio \( A : (A + C) \):
\[ A : (A + C) = A : \frac{19}{9}A = 1 : \frac{19}{9} = 9 : 19 \]

### Final Answer:
Thus, the ratio \( A : (A + C) \) is:
\[ \boxed{9 : 19} \]