If ` (2 + a)/( a) + (2 + b)/( b) + (2 + c)/(c) = 4 ` then the value of `((a b + bc + ca)/(abc)) ` is

Options

A. (a) 2
B. (b) 1
C. (c) 0
D. (d) `(1)/(2)`

Answer

To solve the equation given in the problem, we start with the equation:

\[ \frac{2 + a}{a} + \frac{2 + b}{b} + \frac{2 + c}{c} = 4 \]

### Step 1: Rewrite the fractions
We can break down each term in the equation:

\[ \frac{2 + a}{a} = \frac{2}{a} + \frac{a}{a} = \frac{2}{a} + 1 \]

So, we can rewrite the entire equation as:

\[ \left(\frac{2}{a} + 1\right) + \left(\frac{2}{b} + 1\right) + \left(\frac{2}{c} + 1\right) = 4 \]

### Step 2: Combine like terms
Now, combining the constant terms gives us:

\[ \frac{2}{a} + \frac{2}{b} + \frac{2}{c} + 3 = 4 \]

### Step 3: Isolate the fraction terms
Subtract 3 from both sides:

\[ \frac{2}{a} + \frac{2}{b} + \frac{2}{c} = 1 \]

### Step 4: Factor out the 2
We can factor out the 2 from the left side:

\[ 2\left(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) = 1 \]

### Step 5: Divide both sides by 2
Dividing both sides by 2 gives:

\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{2} \]

### Step 6: Find a common denominator
To combine the fractions on the left side, we find a common denominator, which is \(abc\):

\[ \frac{bc + ac + ab}{abc} = \frac{1}{2} \]

### Step 7: Cross-multiply
Cross-multiplying gives:

\[ 2(bc + ac + ab) = abc \]

### Step 8: Rearranging
Now, we can express \(bc + ac + ab\):

\[ bc + ac + ab = \frac{abc}{2} \]

### Step 9: Find the required value
We need to find the value of:

\[ \frac{ab + bc + ca}{abc} \]

Substituting \(ab + bc + ca\) from the previous step:

\[ \frac{ab + bc + ca}{abc} = \frac{\frac{abc}{2}}{abc} = \frac{1}{2} \]

### Final Answer
Thus, the value of \(\frac{ab + bc + ca}{abc}\) is:

\[ \frac{1}{2} \]