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

According to the problem:

1. The difference between the numerator and the denominator is 27: $x - y = 27$
2. When 3 is added to the denominator, the fraction becomes $\frac{1}{4}$: $\frac{x}{y + 3} = \frac{1}{4}$

Step 1: Solve for $x$ in terms of $y$

From the first equation: $x = y + 27$

Step 2: Substitute this expression into the second equation

Substitute $x = y + 27$ into $\frac{x}{y + 3} = \frac{1}{4}$: $\frac{y + 27}{y + 3} = \frac{1}{4}$

Step 3: Cross-multiply and simplify

Cross-multiply to eliminate the fractions: $4(y + 27) = 1(y + 3)$ Simplifying: $4y + 108 = y + 3$ Now, subtract $y$ from both sides: $3y + 108 = 3$ Next, subtract 108 from both sides: $3y = -105$ Divide by 3: $y = -35$

Step 4: Find $x$

Now, substitute $y = -35$ back into the equation $x = y + 27$: $x = -35 + 27 = -8$

Final Answer:

The fraction is $\frac{-8}{-35} = \frac{8}{35}$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How can you check if the fraction $\frac{8}{35}$ satisfies the given conditions?
2. What happens to the fraction if the numerator and denominator are both doubled?
3. Can you find another fraction that satisfies a similar condition with a different difference between numerator and denominator?
4. How would the solution change if the fraction becomes $\frac{1}{5}$ when 3 is added to the denominator?
5. How would you solve the problem if 5 was added to the denominator instead of 3?

Tip: When solving word problems, it's helpful to assign variables for unknowns and translate the conditions into equations step by step.