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Let's break down the problem step by step and solve it:

The problem states:

- The numerator of a fraction is one less than its denominator.
- If 3 is added to both the numerator and the denominator, the fraction increases by \( \frac{3}{28} \).

We are asked to find the fraction.

### Step 1: Define the variables
Let:
- \( x \) be the numerator of the fraction.
- \( x + 1 \) be the denominator of the fraction (since the numerator is one less than the denominator).

The original fraction is:
\[
\frac{x}{x+1}
\]

### Step 2: Form the new fraction
If we add 3 to both the numerator and denominator, the new fraction becomes:
\[
\frac{x + 3}{(x + 1) + 3} = \frac{x + 3}{x + 4}
\]

### Step 3: Use the condition about the increase
We are told that the new fraction is \( \frac{3}{28} \) greater than the original fraction:
\[
\frac{x + 3}{x + 4} = \frac{x}{x + 1} + \frac{3}{28}
\]

### Step 4: Solve the equation
To solve this equation, we need to combine both fractions on the right-hand side and then cross-multiply to eliminate the denominators.

The equation becomes:
\[
\frac{x + 3}{x + 4} = \frac{28x + 3(x + 1)}{28(x + 1)}
\]

Simplify the right-hand side:
\[
\frac{x + 3}{x + 4} = \frac{28x + 3x + 3}{28(x + 1)} = \frac{31x + 3}{28(x + 1)}
\]

Now, cross-multiply:
\[
28(x + 4)(x + 3) = (x + 4)(31x + 3)
\]

Cancel out \( (x + 4) \) from both sides:
\[
28(x + 3) = 31x + 3
\]

Simplify and solve for \( x \):
\[
28x + 84 = 31x + 3
\]
\[
84 - 3 = 31x - 28x
\]
\[
81 = 3x
\]
\[
x = 27
\]

### Step 5: Find the fraction
Now that we know \( x = 27 \), the numerator is 27 and the denominator is \( x + 1 = 28 \).

So, the fraction is:
\[
\frac{27}{28}
\]

Would you like further details or clarification on any of the steps?

---

Here are 5 related questions to explore further:

1. What happens to the fraction if we add 4 instead of 3 to both the numerator and denominator?
2. Can you derive a general formula for the increase in fraction if \( k \) is added to both the numerator and denominator?
3. What would the fraction be if the numerator was two less than the denominator instead of one less?
4. How would the solution change if the increase in the fraction was \( \frac{5}{28} \) instead of \( \frac{3}{28} \)?
5. What if the original fraction were expressed as \( \frac{x}{x+2} \)? How would that affect the final result?

### Tip:
When solving fractional equations, always aim to simplify the problem by eliminating fractions early using multiplication to avoid complex algebra later on.

The numerator of a fraction is one less than its denominator. If 3 is added to its numerator and denominator, the fraction is increased by 3/28. Find the fraction.

Learn how to solve a fraction problem where the numerator is one less than the denominator and adding 3 increases the fraction by 3/28. This solution covers algebraic fraction solving for Grades 8-10.

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