three unequal positive rational numbers a+b+c=1/(a+b+c)

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.

Explore how to solve equations involving three unequal positive rational numbers with a step-by-step solution. This problem covers algebraic concepts and solving rational equations, making it ideal for students in Grades 9-11.

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