Solving Proportions to Find the Ratio x : y

In algebra, finding the ratio of variables from a given proportion can be a common and useful exercise. In this article, we will walk through the steps to solve the proportion (3x 5y : 7x - 4y 7 : 4) and find the ratio (x : y).

Introduction to the Problem

Given the proportion (3x 5y : 7x - 4y 7 : 4), our goal is to find the ratio of (x : y). This can be approached by setting up an equation based on the equality of the ratios.

Setting Up the Equation

Starting from the proportion:
(frac{3x 5y}{7x - 4y} frac{7}{4})

We can cross-multiply to eliminate the fraction:

4(3x 5y) 7(7x - 4y)

Expanding both sides:

12x 20y 49x - 28y

Now, we can rearrange the equation to isolate terms involving (x) and (y):

12x 20y - 28y 49x

This simplifies to:

12x - 8y 49x

Now, move (12x) to the right side:

-8y 37x

To express the ratio (x : y), we rewrite this as:

(frac{x}{y} frac{48}{37})

Thus, the ratio (x : y) is:

boxed{37 : 48}

Steps to Solve Such Problems

Cross-multiply to eliminate the fraction. Collect (x)'s on one side and (y)'s on the other. Isolate the terms involving (x) and (y). Divide both sides by (y). Divide both sides by the coefficient of (x).

Additional Examples and Practice

To further solidify your understanding, consider these additional examples:

3x 5y / 7x - 4y 7/4 or 3x 5y 7/4(7x - 4y)... This simplifies to the same result: (frac{x}{y} frac{48}{37}). 3x 5y / 7x - 4y 7/4, cross multiply gives 12x 20y 49x - 28y, simplifying leads to 37x 48y, thus x/y 48/37. Express (frac{3x 5y}{7x - 4y} frac{7}{4}) then multiply by the denominator to get 4(3x 5y) 7(7x - 4y), simplify to 37y 48x, hence x/y 48/37.

These examples show that the process is consistent and follows the same steps as the initial problem.

Conclusion

Mastering the skill of solving these types of proportions is crucial for many real-world applications in mathematics and engineering. By practicing and understanding the underlying principles of cross-multiplication and variable isolation, you can confidently tackle more complex problems.