We are given the equation involving three unequal positive rational numbers:

$a + b + c = \frac{1}{a + b + c}$

Step 1: Let’s simplify the equation.

Let $S = a + b + c$, then the equation becomes:

$S = \frac{1}{S}$

Step 2: Solve for $S$.

Multiply both sides by $S$ to eliminate the fraction:

$S^2 = 1$

Taking the square root of both sides:

S = 1 \quad \text{(since a + b + c must be positive)}

Step 3: Interpretation

Thus, we have $a + b + c = 1$. However, this equation does not provide specific values for $a$, $b$, and $c$, except that their sum equals 1. Since $a$, $b$, and $c$ must be unequal positive rational numbers, any unequal values that satisfy $a + b + c = 1$ will work.

Example solution:

Let’s choose some unequal rational numbers that sum to 1. For instance:

• $a = \frac{1}{4}$
• $b = \frac{1}{3}$
• $c = \frac{5}{12}$

Check the sum:

$a + b + c = \frac{1}{4} + \frac{1}{3} + \frac{5}{12} = \frac{3}{12} + \frac{4}{12} + \frac{5}{12} = \frac{12}{12} = 1$

Thus, $a = \frac{1}{4}$, $b = \frac{1}{3}$, and $c = \frac{5}{12}$ are valid unequal rational numbers that satisfy the equation $a + b + c = 1$.

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. What are other sets of unequal rational numbers that satisfy $a + b + c = 1$?
2. How would the equation change if we used different types of numbers (irrational, integers, etc.)?
3. Can there be a geometric interpretation of this equation?
4. What happens if we modify the equation to $a + b + c = \frac{2}{a + b + c}$?
5. How would you solve for $a$, $b$, and $c$ if additional constraints were given?

Tip: When solving for multiple variables, always simplify the equation first before plugging in possible values.