A fraction is such that when 9 is added to the numerator, it becomes 2/3. Half the sum of the numerator and denominator is 28. Find the fraction.

Let the fraction be represented as \( \frac{x}{y} \), where \( x \) is the numerator and \( y \) is the denominator.

### Step 1: Set up the equations
1. **When 9 is added to the numerator, the fraction becomes \( \frac{2}{3} \):**
   \[
   \frac{x + 9}{y} = \frac{2}{3}
   \]
   Cross-multiply to clear the fraction:
   \[
   3(x + 9) = 2y
   \]
   Expanding this, we get:
   \[
   3x + 27 = 2y
   \]
   This simplifies to:
   \[
   3x - 2y = -27 \tag{Equation 1}
   \]

2. **Half the sum of the numerator and denominator is 28:**
   \[
   \frac{x + y}{2} = 28
   \]
   Multiply both sides by 2:
   \[
   x + y = 56 \tag{Equation 2}
   \]

### Step 2: Solve the system of equations
Now we have:
\[
3x - 2y = -27
\]
\[
x + y = 56
\]

From Equation 2, we can express \( y \) in terms of \( x \):
\[
y = 56 - x
\]

Substitute \( y = 56 - x \) into Equation 1:
\[
3x - 2(56 - x) = -27
\]
Expanding and simplifying:
\[
3x - 112 + 2x = -27
\]
\[
5x = 85
\]
\[
x = 17
\]

Now substitute \( x = 17 \) back into Equation 2 to find \( y \):
\[
17 + y = 56
\]
\[
y = 39
\]

### Conclusion
The fraction is:
\[
\frac{x}{y} = \frac{17}{39}
\]

Would you like further details or have any questions?

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Here are some related questions:
1. How would you solve if 9 was subtracted instead of added to the numerator?
2. What if the fraction became \( \frac{3}{4} \) when 9 was added?
3. How does the solution change if the half-sum is a different value?
4. What if we know only one equation about the fraction?
5. How can we verify the fraction obtained is correct?

**Tip:** In word problems involving fractions, translating statements into algebraic equations is key to setting up the problem for solving.

Explore how to solve a fraction problem where adding 9 to the numerator yields 2/3 and the half-sum of numerator and denominator is 28. Step-by-step algebraic methods and equation setup guide included, ideal for Grades 7-9.

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