if `p=(8ab)/(a+b)`, then find the value of `((p+4a)/(p-4a)+(p+4b)/(p-4b))`.

Options

A. 4
B. 2
C. 1
D. 3

Explanation

To solve the problem, we need to find the value of the expression \(\frac{p + 4a}{p - 4a} + \frac{p + 4b}{p - 4b}\) given that \(p = \frac{8ab}{a + b}\).

### Step-by-Step Solution:

1. **Substituting the value of \(p\)**:
We know that \(p = \frac{8ab}{a + b}\). We will substitute this into the expression we need to evaluate.

\[ \frac{p + 4a}{p - 4a} + \frac{p + 4b}{p - 4b} = \frac{\frac{8ab}{a+b} + 4a}{\frac{8ab}{a+b} - 4a} + \frac{\frac{8ab}{a+b} + 4b}{\frac{8ab}{a+b} - 4b} \]

2. **Simplifying the first fraction**:
Let's simplify \(\frac{p + 4a}{p - 4a}\):

- The numerator becomes:
\[ \frac{8ab}{a+b} + 4a = \frac{8ab + 4a(a + b)}{a + b} = \frac{8ab + 4a^2 + 4ab}{a + b} = \frac{12ab + 4a^2}{a + b} \]

- The denominator becomes:
\[ \frac{8ab}{a+b} - 4a = \frac{8ab - 4a(a + b)}{a + b} = \frac{8ab - 4a^2 - 4ab}{a + b} = \frac{4ab - 4a^2}{a + b} = \frac{4a(b - a)}{a + b} \]

Therefore, we have:
\[ \frac{p + 4a}{p - 4a} = \frac{12ab + 4a^2}{4a(b - a)} = \frac{3(3ab + a^2)}{a(b - a)} \]

3. **Simplifying the second fraction**:
Now, simplify \(\frac{p + 4b}{p - 4b}\):

- The numerator becomes:
\[ \frac{8ab}{a+b} + 4b = \frac{8ab + 4b(a + b)}{a + b} = \frac{8ab + 4b^2 + 4ab}{a + b} = \frac{12ab + 4b^2}{a + b} \]

- The denominator becomes:
\[ \frac{8ab}{a+b} - 4b = \frac{8ab - 4b(a + b)}{a + b} = \frac{8ab - 4b^2 - 4ab}{a + b} = \frac{4ab - 4b^2}{a + b} = \frac{4b(a - b)}{a + b} \]

Therefore, we have:
\[ \frac{p + 4b}{p - 4b} = \frac{12ab + 4b^2}{4b(a - b)} = \frac{3(3ab + b^2)}{b(a - b)} \]

4. **Combining both fractions**:
Now we can combine both fractions:
\[ \frac{3(3ab + a^2)}{a(b - a)} + \frac{3(3ab + b^2)}{b(a - b)} \]

To combine these, we find a common denominator, which is \(ab(a - b)\):
\[ = \frac{3(3ab + a^2)b}{ab(b - a)} + \frac{3(3ab + b^2)a}{ab(a - b)} \]

This results in:
\[ = \frac{3b(3ab + a^2) - 3a(3ab + b^2)}{ab(b - a)} \]

5. **Simplifying the numerator**:
After simplifying the numerator:
\[ = \frac{3(3ab^2 + a^2b - 3a^2b - ab^2)}{ab(b - a)} = \frac{3(2ab^2 - 2a^2b)}{ab(b - a)} = \frac{6ab(b - a)}{ab(b - a)} = 6 \]

### Final Answer:
Thus, the value of \(\frac{p + 4a}{p - 4a} + \frac{p + 4b}{p - 4b}\) is \(6\).

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