1. **State the problem:** We have a line segment AB with points C and D between A and B. Car X travels \(\frac{3}{4}\) of AB from A to C, and Car Y travels \(\frac{1}{8}\) of BA from B to D. The distance between points C and D is 20 km. We need to find the length of AB. 2. **Set up variables and expressions:** Let the length of AB be \(x\) km. - Since Car X travels \(\frac{3}{4}\) of AB from A, point C is at distance \(\frac{3}{4}x\) from A. - Car Y travels \(\frac{1}{8}\) of BA from B, which means from B towards A. Since BA is the same length as AB, point D is at distance \(\frac{1}{8}x\) from B towards A. 3. **Locate points on the line:** Since A is at 0 and B is at \(x\), - Point C is at \(\frac{3}{4}x\) from A. - Point D is at \(x - \frac{1}{8}x = \frac{7}{8}x\) from A. 4. **Calculate the distance between C and D:** The distance between C and D is $$\left| \frac{7}{8}x - \frac{3}{4}x \right| = \left| \frac{7}{8}x - \frac{6}{8}x \right| = \frac{1}{8}x$$ 5. **Use the given distance:** We know this distance is 20 km, so $$\frac{1}{8}x = 20$$ 6. **Solve for \(x\):** Multiply both sides by 8: $$x = 20 \times 8 = 160$$ 7. **Conclusion:** The distance between A and B is \(\boxed{160}\) km.