If a : b = 2: 5 , then ( 3a+ 4b) : (5a +6b) is equal to

Options

A. `13//20`
B. ` 26//33`
C. ` 16//27`
D. ` 18//35`

Answer

To solve the problem where \( a : b = 2 : 5 \) and we need to find the value of \( (3a + 4b) : (5a + 6b) \), we can follow these steps:

### Step-by-Step Solution:

1. **Convert the Ratio to Fraction**:
Given \( a : b = 2 : 5 \), we can express this as a fraction:
\[ \frac{a}{b} = \frac{2}{5} \]

2. **Express \( a \) in terms of \( b \)**:
From the fraction, we can express \( a \) as:
\[ a = \frac{2}{5}b \]

3. **Substitute \( a \) in the Expression**:
We need to evaluate \( (3a + 4b) : (5a + 6b) \). Substitute \( a \) in both the numerator and denominator:
- Numerator:
\[ 3a + 4b = 3\left(\frac{2}{5}b\right) + 4b = \frac{6}{5}b + 4b \]
- Denominator:
\[ 5a + 6b = 5\left(\frac{2}{5}b\right) + 6b = 2b + 6b \]

4. **Simplify the Numerator and Denominator**:
Now simplify both parts:
- Numerator:
\[ \frac{6}{5}b + 4b = \frac{6}{5}b + \frac{20}{5}b = \frac{26}{5}b \]
- Denominator:
\[ 2b + 6b = 8b \]

5. **Form the New Ratio**:
Now we can form the ratio:
\[ \frac{(3a + 4b)}{(5a + 6b)} = \frac{\frac{26}{5}b}{8b} \]

6. **Cancel \( b \)**:
Since \( b \) is common in both numerator and denominator, we can cancel it out:
\[ = \frac{26/5}{8} = \frac{26}{5 \times 8} = \frac{26}{40} \]

7. **Simplify the Fraction**:
Now simplify \( \frac{26}{40} \):
\[ = \frac{13}{20} \]

### Final Answer:
Thus, the value of \( (3a + 4b) : (5a + 6b) \) is:
\[ \frac{13}{20} \]

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