If `(2a+b)/(a+4b)=3`, the value of `(a+b)/(a+2b)` is equal to

Options

A. `5/9`
B. `2/7`
C. `4/3`
D. `10/9`

Answer

To solve the equation \(\frac{2a+b}{a+4b}=3\) and find the value of \(\frac{a+b}{a+2b}\), we can follow these steps:

### Step 1: Cross Multiply
Start with the equation:
\[ \frac{2a+b}{a+4b} = 3 \]
Cross multiplying gives:
\[ 2a + b = 3(a + 4b) \]

### Step 2: Expand the Right Side
Now, expand the right side:
\[ 2a + b = 3a + 12b \]

### Step 3: Rearrange the Equation
Rearranging the equation to isolate terms involving \(a\) and \(b\):
\[ 2a + b - 3a - 12b = 0 \]
This simplifies to:
\[ -a - 11b = 0 \]

### Step 4: Solve for \(a\) in terms of \(b\)
Rearranging gives:
\[ a = -11b \]

### Step 5: Substitute \(a\) into \(\frac{a+b}{a+2b}\)
Now substitute \(a = -11b\) into the expression \(\frac{a+b}{a+2b}\):
\[ \frac{-11b + b}{-11b + 2b} = \frac{-10b}{-9b} \]

### Step 6: Simplify the Expression
This simplifies to:
\[ \frac{-10b}{-9b} = \frac{10}{9} \]

### Conclusion
Thus, the value of \(\frac{a+b}{a+2b}\) is:
\[ \frac{10}{9} \]