1. **Stating the problem:** Given the ratios $x : y = 0.5 : 1$ and $z : y = 0.6 : 1$, find the combined ratio $x : y : z$ and the ratio $x : z$. 2. **Understanding ratios:** Ratios compare quantities relative to each other. To combine ratios involving a common term, express all parts with respect to that common term. 3. **Given ratios:** - $x : y = 0.5 : 1$ - $z : y = 0.6 : 1$ 4. **Expressing all in terms of $y$:** - From $x : y = 0.5 : 1$, we have $x = 0.5y$ - From $z : y = 0.6 : 1$, we have $z = 0.6y$ 5. **Forming the combined ratio $x : y : z$:** Substitute the values: $$x : y : z = 0.5y : y : 0.6y$$ Divide all terms by $y$ to simplify: $$0.5 : 1 : 0.6$$ 6. **Eliminate decimals for clarity:** Multiply all terms by 10: $$5 : 10 : 6$$ 7. **Simplify the ratio:** Divide all terms by their greatest common divisor, which is 1 here, so the ratio remains: $$5 : 10 : 6$$ 8. **Find the ratio $x : z$:** From the simplified ratio, $x : z = 5 : 6$ **Final answers:** - $x : y : z = 5 : 10 : 6$ - $x : z = 5 : 6$