If 𝑥: 𝑦 = 3: 4, find (4𝑥 + 5𝑦): (5𝑥 − 27)

Given the ratio \( x : y = 3 : 4 \), we can express \( x \) and \( y \) in terms of a constant \( k \) such that:

\[
x = 3k \quad \text{and} \quad y = 4k
\]

We are asked to find the ratio \( (4x + 5y) : (5x - 27) \). Let's substitute the expressions for \( x \) and \( y \) into the ratio:

\[
4x + 5y = 4(3k) + 5(4k) = 12k + 20k = 32k
\]

\[
5x - 27 = 5(3k) - 27 = 15k - 27
\]

Thus, the ratio becomes:

\[
\frac{4x + 5y}{5x - 27} = \frac{32k}{15k - 27}
\]

This is the required ratio in its simplest form. 

If you have further questions or need more details, feel free to ask!

Here are five related questions you might find useful:
1. How would the ratio change if \( x : y = 5 : 7 \)?
2. What happens to the expression \( (4x + 5y) : (5x - 27) \) if \( y \) becomes negative?
3. How do you solve a problem where the ratio \( x : y : z \) is given as 2 : 3 : 5?
4. How do you simplify ratios involving more complex algebraic expressions?
5. What are some real-world applications of solving ratios like these?

**Tip:** When working with ratios, always try to express the variables in terms of a single constant. This simplifies calculations and helps avoid errors.

Learn how to solve the ratio problem involving the ratios x : y = 3 : 4. Find the ratio (4x + 5y) : (5x - 27) using algebraic expressions and substitution. Suitable for high school students and those interested in ratio and proportion problems.

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