If `a/b=3/4` then find the value of `7a-4b:3a+b`.

Options

A. `7:1`
B. `5:13`
C. `12:1`
D. None of these

Answer

To solve the problem, we need to find the value of the ratio \( \frac{7a - 4b}{3a + b} \) given that \( \frac{a}{b} = \frac{3}{4} \).

### Step-by-step Solution:

1. **Express \( a \) in terms of \( b \)**:
Given \( \frac{a}{b} = \frac{3}{4} \), we can express \( a \) as:
\[ a = \frac{3}{4}b \]

2. **Substitute \( a \) into the expression \( 7a - 4b \)**:
Substitute \( a = \frac{3}{4}b \) into \( 7a - 4b \):
\[ 7a - 4b = 7\left(\frac{3}{4}b\right) - 4b \]
Simplifying this gives:
\[ = \frac{21}{4}b - 4b = \frac{21}{4}b - \frac{16}{4}b = \frac{5}{4}b \]

3. **Substitute \( a \) into the expression \( 3a + b \)**:
Now substitute \( a = \frac{3}{4}b \) into \( 3a + b \):
\[ 3a + b = 3\left(\frac{3}{4}b\right) + b \]
Simplifying this gives:
\[ = \frac{9}{4}b + b = \frac{9}{4}b + \frac{4}{4}b = \frac{13}{4}b \]

4. **Form the ratio \( \frac{7a - 4b}{3a + b} \)**:
Now we can form the ratio:
\[ \frac{7a - 4b}{3a + b} = \frac{\frac{5}{4}b}{\frac{13}{4}b} \]
The \( b \) cancels out (assuming \( b \neq 0 \)):
\[ = \frac{5}{13} \]

5. **Final Result**:
Therefore, the value of \( 7a - 4b : 3a + b \) is:
\[ 5 : 13 \]