If `(2 a + b)/( a + 4b) = 3 ` then find the value of `(a + b)/( a + 2b)`

Options

A. `(5)/(9)`
B. `(2)/(7)`
C. `(10)/(9)`
D. `(10)/(7)`

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
Starting with the given equation:
\[ \frac{2a + b}{a + 4b} = 3 \]
We can cross-multiply:
\[ 2a + b = 3(a + 4b) \]

### Step 2: Distribute on the Right Side
Distributing the 3 on the right side:
\[ 2a + b = 3a + 12b \]

### Step 3: Rearrange the Equation
Now, we will rearrange the equation to isolate terms involving \(a\) and \(b\):
\[ 2a - 3a = 12b - b \]
This simplifies to:
\[ -a = 11b \]

### Step 4: Solve for \(a\)
From the equation \(-a = 11b\), we can solve for \(a\):
\[ a = -11b \]

### Step 5: Substitute \(a\) into the Second Expression
Now we need to find the value of \(\frac{a + b}{a + 2b}\). Substituting \(a = -11b\) into this expression:
\[ \frac{-11b + b}{-11b + 2b} \]

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

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

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