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Let's analyze the equation:

\[
\frac{16a^2 + 15b}{20ab} = \frac{Xa}{Yb} + \frac{3}{Za}
\]

We are tasked with finding the values of \( X \), \( Y \), and \( Z \), which are the smallest positive integers that satisfy this equation.

### Step 1: Simplifying the Left-Hand Side (LHS)
The left-hand side of the equation is:

\[
\frac{16a^2 + 15b}{20ab}
\]

We will try to split this into two fractions so that it matches the form on the right-hand side of the equation. First, express the numerator as the sum of two terms:

\[
\frac{16a^2}{20ab} + \frac{15b}{20ab}
\]

Simplifying each term separately:

\[
\frac{16a^2}{20ab} = \frac{4a}{5b}, \quad \frac{15b}{20ab} = \frac{3}{4a}
\]

So, the left-hand side becomes:

\[
\frac{4a}{5b} + \frac{3}{4a}
\]

### Step 2: Comparing with the Right-Hand Side (RHS)
The right-hand side of the equation is:

\[
\frac{Xa}{Yb} + \frac{3}{Za}
\]

From this comparison, we can deduce:

\[
\frac{4a}{5b} + \frac{3}{4a} = \frac{Xa}{Yb} + \frac{3}{Za}
\]

This implies that:
\[
X = 4, \quad Y = 5, \quad Z = 4
\]

### Final Values:
- \( X = 4 \)
- \( Y = 5 \)
- \( Z = 4 \)

Would you like further clarification or more details? 

Here are 5 related questions for further practice:
1. How would you solve a similar equation if the coefficients were different?
2. Can you solve a similar equation where the variables are on both sides?
3. How would you handle this equation if the fraction were more complex?
4. Can you create a similar problem but with different powers of \( a \) and \( b \)?
5. How would the solution change if one of the pronumerals were squared?

**Tip:** When dealing with rational expressions, breaking them into simpler fractions can make solving easier.

Correctly complete the equation by finding the values of the capitalized pronumerals. Your answers should be the lowest positive numbers possible. (16a^2 + 15b) / (20ab) = Xa / Yb + 3 / Za

Learn how to solve an algebraic equation involving rational expressions by finding the lowest positive values for the pronumerals X, Y, and Z. Detailed step-by-step simplification is provided. Suitable for students in Grades 9-10.

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