If ` a + b + c = 5 ,ab + bc + ca = 7 and abc = 3 ` find the value of ` ((a)/( b) + (b)/( a)) + (( b)/( c) + (c)/( b)) + ((c)/(a) + (a)/( c))`

Options

A. `8 (2)/(3)`
B. `7 (2)/(3)`
C. `9 (2)/(3)`
D. `8 (1)/(3)`

Answer

To solve the problem, we need to find the value of the expression:

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

Given the equations:

1. \( a + b + c = 5 \)
2. \( ab + bc + ca = 7 \)
3. \( abc = 3 \)

### Step 1: Rewrite the expression

We can rewrite the expression using the identity:

\[ \frac{x}{y} + \frac{y}{x} = \frac{x^2 + y^2}{xy} \]

Thus, we can express our target expression as:

\[ \left( \frac{a^2 + b^2}{ab} \right) + \left( \frac{b^2 + c^2}{bc} \right) + \left( \frac{c^2 + a^2}{ca} \right) \]

### Step 2: Calculate \( a^2 + b^2 + c^2 \)

Using the identity \( a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + ac + bc) \):

\[ a^2 + b^2 + c^2 = 5^2 - 2 \cdot 7 = 25 - 14 = 11 \]

### Step 3: Calculate \( ab + ac + bc \)

From the problem, we already know:

\[ ab + ac + bc = 7 \]

### Step 4: Calculate \( \frac{a^2 + b^2}{ab} + \frac{b^2 + c^2}{bc} + \frac{c^2 + a^2}{ca} \)

Now, we can express \( a^2 + b^2 \) as:

\[ a^2 + b^2 = (a + b)^2 - 2ab \]

Using \( a + b = 5 - c \):

\[ a^2 + b^2 = (5 - c)^2 - 2ab \]

We can express \( a^2 + b^2 + c^2 \) in terms of \( ab \), \( ac \), and \( bc \):

\[ \frac{a^2 + b^2}{ab} = \frac{(5 - c)^2 - 2ab}{ab} \]

### Step 5: Combine all parts

Now, we can substitute \( a^2 + b^2 + c^2 = 11 \) into the expression:

\[ \frac{a^2 + b^2 + c^2}{ab + ac + bc} = \frac{11}{7} \]

Thus, the complete expression simplifies to:

\[ \frac{11}{ab} + \frac{11}{bc} + \frac{11}{ca} \]

### Step 6: Calculate the final value

The final value can be calculated as:

\[ \frac{11}{3} + \frac{11}{3} + \frac{11}{3} = 3 \cdot \frac{11}{3} = 11 \]

### Final Answer

Thus, the value of the expression is:

\[ \boxed{11} \]