If ` (a)/( b) = (2)/(3) and (b)/( c) = (4)/( 5)` , then the ratio `(a + b)/(b + c) ` equal to

Options

A. `(20)/(27)`
B. `(27)/(20)`
C. `(6)/(8)`
D. `(8)/(6)`

Answer

To solve the problem, we need to find the ratio \((a + b)/(b + c)\) given the ratios \((a/b) = (2/3)\) and \((b/c) = (4/5)\).

### Step-by-Step Solution:

1. **Set Up the Ratios**:
We are given:
\[ \frac{a}{b} = \frac{2}{3} \quad \text{and} \quad \frac{b}{c} = \frac{4}{5} \]

2. **Express \(a\) and \(c\) in terms of \(b\)**:
From the first ratio, we can express \(a\) in terms of \(b\):
\[ a = \frac{2}{3}b \]
From the second ratio, we can express \(c\) in terms of \(b\):
\[ c = \frac{5}{4}b \]

3. **Substitute \(a\) and \(c\) into the expression \((a + b)/(b + c)\)**:
Now we substitute \(a\) and \(c\) into the expression:
\[ \frac{a + b}{b + c} = \frac{\frac{2}{3}b + b}{b + \frac{5}{4}b} \]

4. **Simplify the Numerator**:
The numerator becomes:
\[ \frac{2}{3}b + b = \frac{2}{3}b + \frac{3}{3}b = \frac{5}{3}b \]

5. **Simplify the Denominator**:
The denominator becomes:
\[ b + \frac{5}{4}b = \frac{4}{4}b + \frac{5}{4}b = \frac{9}{4}b \]

6. **Combine the Results**:
Now we have:
\[ \frac{a + b}{b + c} = \frac{\frac{5}{3}b}{\frac{9}{4}b} \]
The \(b\) cancels out:
\[ = \frac{5}{3} \times \frac{4}{9} = \frac{20}{27} \]

7. **Final Result**:
Therefore, the ratio \((a + b)/(b + c)\) is:
\[ \frac{20}{27} \]