If `a : b = 3: 8`, find the value of `(4a + 3b)/(6a-b)`.

Answer

To solve the problem where \( a : b = 3 : 8 \) and we need to find the value of \( \frac{4a + 3b}{6a - b} \), we can follow these steps:

### Step 1: Express \( a \) in terms of \( b \)

Given the ratio \( a : b = 3 : 8 \), we can express \( a \) in terms of \( b \):
\[ a = \frac{3}{8}b \]

### Step 2: Substitute \( a \) in the expression

Now, substitute \( a \) in the expression \( \frac{4a + 3b}{6a - b} \):
\[ \frac{4\left(\frac{3}{8}b\right) + 3b}{6\left(\frac{3}{8}b\right) - b} \]

### Step 3: Simplify the numerator

Calculate the numerator:
\[ 4\left(\frac{3}{8}b\right) + 3b = \frac{12}{8}b + 3b = \frac{12}{8}b + \frac{24}{8}b = \frac{36}{8}b = \frac{9}{2}b \]

### Step 4: Simplify the denominator

Now, calculate the denominator:
\[ 6\left(\frac{3}{8}b\right) - b = \frac{18}{8}b - b = \frac{18}{8}b - \frac{8}{8}b = \frac{10}{8}b = \frac{5}{4}b \]

### Step 5: Form the fraction

Now we can form the fraction:
\[ \frac{\frac{9}{2}b}{\frac{5}{4}b} \]

### Step 6: Cancel \( b \) and simplify

Since \( b \) is common in both the numerator and the denominator, we can cancel it out:
\[ \frac{\frac{9}{2}}{\frac{5}{4}} = \frac{9}{2} \times \frac{4}{5} = \frac{36}{10} = \frac{18}{5} \]

### Final Answer

Thus, the value of \( \frac{4a + 3b}{6a - b} \) is:
\[ \frac{18}{5} \]

---