Shortcut Trick Since B is common in both ratios and has the same value (6), we can directly take A = 5, B = 6, and C = 7. (A + B) = 5 + 6 = 11 (B + C) = 6 + 7 = 13 (C + A) = 7 + 5 = 12 The required ratio is 11 ∶ 13 ∶ 12. ∴ The correct answer is 11 ∶ 13 ∶ 12. Alternate Method Given: A ∶ B = 5 ∶ 6 B ∶ C = 6 ∶ 7 Formula Used: If A ∶ B = a ∶ b and B ∶ C = b ∶ c, then A ∶ B ∶ C = a ∶ b ∶ c. Calculations: ⇒ Since the value of B is 6 in both ratios, A ∶ B ∶ C = 5 ∶ 6 ∶ 7. ⇒ Let A = 5k, B = 6k, and C = 7k. ⇒ (A + B) = 5k + 6k = 11k ⇒ (B + C) = 6k + 7k = 13k ⇒ (C + A) = 7k + 5k = 12k ⇒ (A + B) ∶ (B + C) ∶ (C + A) = 11k ∶ 13k ∶ 12k ⇒ (A + B) ∶ (B + C) ∶ (C + A) = 11 ∶ 13 ∶ 12 ∴ The correct answer is 11 ∶ 13 ∶ 12. Related Concepts: Compound Ratio The compound ratio of (a ∶ b) and (c ∶ d) is (ac ∶ bd). Duplicate Ratio The duplicate ratio of (a ∶ b) is (a^(2) ∶ b^(2)). Combining Ratios To combine A ∶ B and B ∶ C, multiply the ratios such that the term for B becomes equal in both.