If A: B= 4:5 and A: C = 2: 3, then what is `(A+B): (B+C)` ?

Options

A. `9:11`
B. `10:9`
C. `9:10`
D. `11:9`

Answer

To solve the problem, we need to find the ratio \((A + B) : (B + C)\) given the ratios \(A : B = 4 : 5\) and \(A : C = 2 : 3\).

### Step-by-Step Solution:

1. **Express A, B, and C in terms of a common variable:**
- From the ratio \(A : B = 4 : 5\), we can express \(A\) and \(B\) as:
\[ A = 4x \quad \text{and} \quad B = 5x \]
- From the ratio \(A : C = 2 : 3\), we can express \(A\) and \(C\) as:
\[ A = 2y \quad \text{and} \quad C = 3y \]

2. **Equate the expressions for A:**
- Since both expressions represent \(A\), we can set them equal to each other:
\[ 4x = 2y \]
- From this equation, we can express \(y\) in terms of \(x\):
\[ y = 2x \]

3. **Substitute \(y\) back to find C:**
- Now substitute \(y = 2x\) into the expression for \(C\):
\[ C = 3y = 3(2x) = 6x \]

4. **Now we have all variables in terms of \(x\):**
- \(A = 4x\)
- \(B = 5x\)
- \(C = 6x\)

5. **Calculate \(A + B\) and \(B + C\):**
- Calculate \(A + B\):
\[ A + B = 4x + 5x = 9x \]
- Calculate \(B + C\):
\[ B + C = 5x + 6x = 11x \]

6. **Form the ratio \((A + B) : (B + C)\):**
- Now we can form the ratio:
\[ (A + B) : (B + C) = 9x : 11x \]
- This simplifies to:
\[ 9 : 11 \]

### Final Answer:
The ratio \((A + B) : (B + C)\) is \(9 : 11\).