The Mystery of Fractions with Denominators Three More than the Numerator

In mathematics, fractions play a crucial role in various calculations and problem-solving scenarios. A specific type of fraction, where the denominator is exactly three more than the numerator, can often lead to interesting and solvable problems. This article explores how to find such fractions and delves into a detailed solution for the problem: What is the fraction whose denominator is 3 more than the numerator?

Solving the Problem

Let's consider the following scenario: What is the fraction whose denominator is 3 more than the numerator? Given the equations:
(d n 3), where (d) is the denominator and (n) is the numerator.
Additionally, we have some fraction (frac{n 1}{d}) which simplifies to a known ratio, such as (frac{3}{4}).

Step-by-Step Solution

1. First, we recognize that (d n 3) from the equation provided. 2. Applying cross multiplication to (frac{n 1}{d} frac{3}{4}) produces the following equation after simplification: [ 4(n 1) 3d ] 3. Substitute (d n 3) into the simplified equation: [ 4(n 1) 3(n 3) ] 4. Expand and simplify the equation: [ 4n 4 3n 9 ] 5. Isolate (n) by solving the equation: [ n 5 ] 6. Substitute (n 5) back into the equation for (d): [ d n 3 5 3 8 ] 7. Therefore, the fraction is (frac{n}{d} frac{5}{8}).

Additional Examples and Variations

Example 1: Consider the fraction (frac{x}{x 3}). If we add 1 to the numerator and the denominator, the resulting fraction is (frac{x 1}{x 4}). Given (frac{x 1}{x 4} frac{3}{4}), find the original fraction.

Solving the equation:

[frac{x 1}{x 4} frac{3}{4} ][ 4(x 1) 3(x 4) ][ 4x 4 3x 12 ][ x 8 ] [text{Therefore, the original fraction is } frac{8}{11}.]

Example 2: The original fraction is (frac{x}{x 3}). After adding 1 to both the numerator and denominator, the result is (frac{x 1}{x 4} frac{3}{4}). Find the value of x.

[frac{x 1}{x 4} frac{3}{4} ][ 4(x 1) 3(x 4) ][ 4x 4 3x 12 ][ x 8 ] [text{So the original fraction is } frac{8}{11}.]

Conclusion

Through these examples, we now understand how to find a fraction where the denominator is three more than the numerator. Our conclusion is that the fraction (frac{8}{11}) satisfies the given conditions when the numerator and denominator are increased by 1. This is a fascinating property of fractions and demonstrates the power of algebraic manipulation and problem-solving techniques in mathematics.

For more problems and solutions in fractions, please visit our Fractions Page, where you can find a wide range of examples and solutions tailored to your needs.