1. **Problem statement:** We have a straight line WXYZ passing through the centers of two circles. Given the ratios: - $WX : XY = 7 : 2$ - $XY : YZ = 3 : 10$ We need to find the ratio of the diameter of the smaller circle to the diameter of the larger circle. 2. **Understanding the problem:** The points W, X, Y, Z lie on the line through the centers of the two circles. The circles overlap, so the distance between their centers corresponds to segment XY. 3. **Express lengths in terms of a common variable:** From $WX : XY = 7 : 2$, let $WX = 7k$ and $XY = 2k$. From $XY : YZ = 3 : 10$, let $XY = 3m$ and $YZ = 10m$. 4. **Find the common length for $XY$:** Since $XY$ is the same segment, equate $2k = 3m$. Solve for $k$ in terms of $m$: $$2k = 3m \implies k = \frac{3m}{2}$$ 5. **Express all segments in terms of $m$:** $$WX = 7k = 7 \times \frac{3m}{2} = \frac{21m}{2}$$ $$XY = 3m$$ $$YZ = 10m$$ 6. **Calculate total length $WZ$:** $$WZ = WX + XY + YZ = \frac{21m}{2} + 3m + 10m = \frac{21m}{2} + 13m = \frac{21m}{2} + \frac{26m}{2} = \frac{47m}{2}$$ 7. **Relate the diameters to the segments:** The two circles have centers at points $X$ and $Y$. The diameter of the smaller circle is $WX + XY$ or $XY + YZ$ depending on which circle is smaller. Since $WX = \frac{21m}{2} = 10.5m$ and $YZ = 10m$, $WX$ is larger than $YZ$, so the smaller circle is the one with diameter $YZ + XY$. Diameters: - Smaller circle diameter = $XY + YZ = 3m + 10m = 13m$ - Larger circle diameter = $WX + XY = \frac{21m}{2} + 3m = 10.5m + 3m = 13.5m$ 8. **Calculate the ratio of smaller to larger diameter:** $$\frac{13m}{13.5m} = \frac{13}{13.5} = \frac{13}{\frac{27}{2}} = \frac{13 \times 2}{27} = \frac{26}{27}$$ 9. **Simplify the ratio:** $26$ and $27$ have no common factors other than 1, so the ratio in simplest form is: $$\boxed{26 : 27}$$