If `A: B = 1/2 : 1/3, B: C = 1/5 : 1/3`, then (A+B): (B+C) is equal to:

Options

A. `5:8`
B. `9:10`
C. `15:16`
D. `6:15`

Answer

To solve the problem, we need to find the ratio of (A + B) to (B + C) given the ratios A:B and B:C.

### Step-by-Step Solution:

1. **Understand the given ratios**:
- We have A:B = 1/2 : 1/3.
- We also have B:C = 1/5 : 1/3.

2. **Convert the ratios to a more manageable form**:
- For A:B = 1/2 : 1/3, we can cross-multiply to find a common ratio:
\[ A : B = \frac{1/2}{1/3} = \frac{1 \cdot 3}{2 \cdot 1} = \frac{3}{2} \]
- Thus, A:B = 3:2.

3. **For B:C = 1/5 : 1/3, we do the same**:
- Cross-multiply:
\[ B : C = \frac{1/5}{1/3} = \frac{1 \cdot 3}{5 \cdot 1} = \frac{3}{5} \]
- Thus, B:C = 3:5.

4. **Set up the ratios**:
- We now have:
- A:B = 3:2
- B:C = 3:5
- Let’s denote A = 3x, B = 2x, and C = 5y (where x and y are some constants).

5. **Find a common value for B**:
- From A:B = 3:2, we have B = 2x.
- From B:C = 3:5, we have B = 3y.
- Setting these equal gives us:
\[ 2x = 3y \implies y = \frac{2x}{3} \]

6. **Substituting back to find C**:
- Now substitute y back into C:
\[ C = 5y = 5 \left(\frac{2x}{3}\right) = \frac{10x}{3} \]

7. **Now we have A, B, and C in terms of x**:
- A = 3x
- B = 2x
- C = \(\frac{10x}{3}\)

8. **Calculate A + B and B + C**:
- A + B = 3x + 2x = 5x
- B + C = 2x + \frac{10x}{3} = \frac{6x}{3} + \frac{10x}{3} = \frac{16x}{3}

9. **Find the ratio (A + B):(B + C)**:
- We need to find:
\[ (A + B) : (B + C) = 5x : \frac{16x}{3} \]
- To simplify, multiply both sides by 3:
\[ 5x \cdot 3 : 16x = 15 : 16 \]

### Final Answer:
Thus, the ratio (A + B) : (B + C) is equal to **15 : 16**.