Let's solve this problem step by step:

Given:

• The denominator of a fraction is 14 more than the numerator.
• If both numerator and denominator are increased by 5, the fraction becomes $\frac{11}{18}$.

Step 1: Define Variables

Let the original numerator be $x$. Then, the original denominator will be $x + 14$.

Step 2: Create the Equation

When both the numerator and denominator are increased by 5, the new fraction becomes $\frac{11}{18}$.

• New numerator = $x + 5$
• New denominator = $x + 14 + 5 = x + 19$

So, the new fraction is: $\frac{x + 5}{x + 19} = \frac{11}{18}$

Step 3: Cross-Multiply

Cross-multiplying gives: $18(x + 5) = 11(x + 19)$

Step 4: Expand and Simplify

Expanding both sides: $18x + 90 = 11x + 209$

Step 5: Solve for $x$

Subtract $11x$ from both sides: $7x + 90 = 209$ Subtract 90 from both sides: $7x = 119$ Divide by 7: $x = 17$

Step 6: Find the Original Fraction

• Original numerator = $x = 17$
• Original denominator = $x + 14 = 17 + 14 = 31$

So, the original fraction is: $\frac{17}{31}$

Final Answer:

The original fraction is $\frac{17}{31}$.

Would you like further details or have any questions about the solution?

Related Questions:

1. How do you verify the solution for a fraction word problem?
2. What if the difference between the numerator and the denominator was different?
3. How can you solve similar problems where the relationship between numerator and denominator is given?
4. How does cross-multiplying help solve equations involving fractions?
5. What are some common mistakes when setting up equations in word problems?

Tip:

When solving word problems, carefully define your variables and set up the relationships before jumping into the calculations—this makes solving much more manageable!