1. **State the problem:** We are given three ratios involving four sizes $X_1, X_2, X_3,$ and $X_4$: $$X_1 : X_2 = 2 : 3$$ $$X_2 : X_3 = 2 : 3$$ $$X_3 : X_4 = 9 : 17$$ We need to find which of the given four-term ratios is consistent with these relationships. 2. **Understand the problem:** To combine these ratios into one four-term ratio $X_1 : X_2 : X_3 : X_4$, the intermediate terms must be consistent. We will express all terms relative to $X_1$ or $X_2$ and check the options. 3. **Use the given ratios:** - From $X_1 : X_2 = 2 : 3$, we can write $X_1 = 2k$ and $X_2 = 3k$ for some $k$. - From $X_2 : X_3 = 2 : 3$, write $X_2 = 2m$ and $X_3 = 3m$ for some $m$. - From $X_3 : X_4 = 9 : 17$, write $X_3 = 9n$ and $X_4 = 17n$ for some $n$. 4. **Make $X_2$ consistent:** We have $X_2 = 3k$ and $X_2 = 2m$, so set $3k = 2m$. Solve for $m$: $$m = \frac{3k}{2}$$ 5. **Express $X_3$ in terms of $k$:** Since $X_3 = 3m$, substitute $m$: $$X_3 = 3 \times \frac{3k}{2} = \frac{9k}{2}$$ 6. **Make $X_3$ consistent:** We also have $X_3 = 9n$, so set: $$9n = \frac{9k}{2}$$ Solve for $n$: $$n = \frac{k}{2}$$ 7. **Express $X_4$ in terms of $k$:** Since $X_4 = 17n$, substitute $n$: $$X_4 = 17 \times \frac{k}{2} = \frac{17k}{2}$$ 8. **Write all terms in terms of $k$:** $$X_1 = 2k$$ $$X_2 = 3k$$ $$X_3 = \frac{9k}{2}$$ $$X_4 = \frac{17k}{2}$$ 9. **Clear denominators to get integer ratios:** Multiply all terms by 2: $$2 \times X_1 = 4k$$ $$2 \times X_2 = 6k$$ $$2 \times X_3 = 9k$$ $$2 \times X_4 = 17k$$ So the ratio is: $$X_1 : X_2 : X_3 : X_4 = 4 : 6 : 9 : 17$$ 10. **Check the options:** - $2 : 3 : 9 : 17$ (does not match) - $4 : 6 : 18 : 34$ (not matching $X_3$ and $X_4$) - $4 : 6 : 8 : 16$ (does not match) - $4 : 6 : 9 : 17$ (matches exactly) **Final answer:** $$\boxed{4 : 6 : 9 : 17}$$