We are given two ratios:

1. $\frac{x}{y} = \frac{3}{4}$
2. $\frac{y}{z} = \frac{5}{6}$

We want to find the ratio of $\frac{x}{z}$.

Step 1: Express $x$ in terms of $y$

From the first ratio, we can write: $x = \frac{3}{4}y$

Step 2: Express $y$ in terms of $z$

From the second ratio, we can write: $y = \frac{5}{6}z$

Step 3: Substitute $y$ into the equation for $x$

Now, substitute $y = \frac{5}{6}z$ into the equation $x = \frac{3}{4}y$: $x = \frac{3}{4} \times \frac{5}{6}z$

Step 4: Simplify the expression

Multiplying the fractions: $x = \frac{3 \times 5}{4 \times 6}z = \frac{15}{24}z$ Simplify $\frac{15}{24}$: $x = \frac{5}{8}z$

Step 5: Write the ratio of $x$ to $z$

Thus, the ratio of $x$ to $z$ is: $\frac{x}{z} = \frac{5}{8}$

The ratio of $x$ to $z$ is $5:8$.

Would you like a further breakdown of any of the steps? Here are five related questions to explore this further:

1. How do you simplify fractions like $\frac{15}{24}$?
2. What if the ratio of $x$ to $y$ was different? How would that change the result?
3. Can you apply this method to three variables with different ratios?
4. How would you solve if given only the final ratio $\frac{x}{z}$?
5. What does it mean when ratios are proportional across three variables?

Tip: When working with ratios, always make sure units and variables are consistent before simplifying.