1. **State the problem:** We have a road divided into three parts: A, B, and C. - Length of A is $\frac{5}{8}$ of B. - Length of C is 8 km longer than $\frac{5}{6}$ of B. - Total length of the road is 67 km. Find the lengths of A, B, and C. 2. **Set variables:** Let the length of part B be $x$ km. Then: - $A = \frac{5}{8}x$ - $C = \frac{5}{6}x + 8$ 3. **Write the total length equation:** $$A + B + C = 67$$ Substitute the expressions: $$\frac{5}{8}x + x + \left(\frac{5}{6}x + 8\right) = 67$$ 4. **Combine like terms:** $$\frac{5}{8}x + x + \frac{5}{6}x + 8 = 67$$ Convert all terms to have a common denominator (24): $$\frac{15}{24}x + \frac{24}{24}x + \frac{20}{24}x + 8 = 67$$ Sum the fractions: $$\left(\frac{15 + 24 + 20}{24}\right)x + 8 = 67$$ $$\frac{59}{24}x + 8 = 67$$ 5. **Isolate $x$:** $$\frac{59}{24}x = 67 - 8$$ $$\frac{59}{24}x = 59$$ 6. **Solve for $x$:** $$x = \frac{59}{1} \times \frac{24}{59} = 24$$ 7. **Find lengths of A and C:** $$A = \frac{5}{8} \times 24 = 15$$ $$C = \frac{5}{6} \times 24 + 8 = 20 + 8 = 28$$ 8. **Check total:** $$15 + 24 + 28 = 67$$ which matches the total length. **Final answer:** - Length of A is 15 km - Length of B is 24 km - Length of C is 28 km