To solve the problem, let's start by defining the given conditions clearly: $a + b + c = 11$ $\frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{a+c} = \frac{13}{17}$

We are asked to find the value of: $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b}$

To proceed, we utilize the identity for sums of fractions and algebraic manipulation. First, recall a useful identity in algebra which can help solve this:

For any three numbers $a, b, c$, $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = \frac{3}{2} \left( 1 + \frac{\frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{a+c}}{a + b + c} \right)$

Given: $a + b + c = 11$ $\frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{a+c} = \frac{13}{17}$

Let's substitute these values into the identity: $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = \frac{3}{2} \left( 1 + \frac{\frac{13}{17}}{11} \right)$

Simplify inside the parentheses first: $1 + \frac{\frac{13}{17}}{11} = 1 + \frac{13}{187}$ $1 + \frac{13}{187} = \frac{187}{187} + \frac{13}{187} = \frac{200}{187}$

Now, substitute back into the formula: $\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = \frac{3}{2} \left( \frac{200}{187} \right)$ $= \frac{3 \times 200}{2 \times 187}$ $= \frac{600}{374}$ $= \frac{300}{187}$

So, the final answer is: $\boxed{\frac{300}{187}}$

Would you like any details or have any questions?

Here are 5 questions you might be interested in:

1. How can I solve systems of equations involving three variables?
2. What are some common algebraic identities useful in problem-solving?
3. How do I approach solving complex fractions in equations?
4. Can you explain more about the identity used in this problem?
5. What techniques can help simplify algebraic expressions?

Tip: When working with systems of equations, always check for symmetry or known algebraic identities that might simplify the problem.