Let the fraction be $\frac{a}{b}$, where $a$ is the numerator and $b$ is the denominator.

Step 1: Translate the problem into equations

1. The sum of the numerator and denominator is 17: a + b = 17 \tag{1}

2. If 3 is added to the numerator, the value of the fraction becomes 1: \frac{a + 3}{b} = 1 \tag{2}

Step 2: Solve the equations

From equation (2), simplify: $a + 3 = b$ b = a + 3 \tag{3}

Substitute $b = a + 3$ into equation (1): $a + (a + 3) = 17$ $2a + 3 = 17$ $2a = 14$ $a = 7$

Substitute $a = 7$ into equation (3): $b = 7 + 3 = 10$

Step 3: Verify the solution

The fraction is $\frac{7}{10}$. Check:

1. $7 + 10 = 17$ (satisfied).
2. Adding 3 to the numerator gives $\frac{7 + 3}{10} = \frac{10}{10} = 1$ (satisfied).

Thus, the fraction is: $\boxed{\frac{7}{10}}$

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Do you want a detailed explanation of any step, or have further questions?
Here are 5 related questions to explore:

1. How can we find the simplest form of a fraction?
2. What if the sum of the numerator and denominator was a different number?
3. How would the solution change if the fraction equaled 2 instead of 1 after the numerator increased?
4. What other methods can we use to solve such problems?
5. Can fractions always be expressed in their simplest form using similar approaches?

Tip: Always double-check your solution by verifying it against all given conditions!