if a+b+c=11, and (1/[a+b])+(1/[b+c])+(1/[a+c])=(13/17), what is (a/[b+c])+(b/[a+c])+(c/[a+b]) equal to?

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.

Explore how to solve equations involving fractions with a detailed step-by-step solution. This problem covers key algebraic concepts such as equations and fractions, using the sum of fractions identity. Suitable for students in Grades 10-12.

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