The difference between the numerator and the denominator of the fraction is 27. when 3 is added to the denominator, the fraction becomes 1/4. What is the fraction.

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.

The difference between the numerator and the denominator of the fraction is 27. When 3 is added to the denominator, the fraction becomes 1/4. What is the fraction?

Learn how to solve a fraction problem where the difference between the numerator and denominator is 27. When 3 is added to the denominator, the fraction becomes 1/4. This step-by-step solution involves cross-multiplication and substitution, making it ideal for students in Grades 6-8.

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