Linked equations to combined ratio: If 4a = 5b and 7b = 9c, find the simplified ratio a : b : c.

Introduction / Context: Converting linear relations (like 4a = 5b and 7b = 9c) into a three-term ratio a : b : c requires aligning the common variable b and then computing consistent values for a and c. This is a standard algebraic technique for ratio building.

Given Data / Assumptions:

• 4a = 5b.
• 7b = 9c.
• All terms are positive, allowing ratio construction.

Concept / Approach: From 4a = 5b → a : b = 5 : 4. From 7b = 9c → b : c = 9 : 7. Combine by making b equal in both pairs and then read off a, b, c.

Step-by-Step Solution: From 4a = 5b → a : b = 5 : 4. From 7b = 9c → b : c = 9 : 7. Equalize b: choose b = LCM(4, 9) = 36. If b = 36, then from a : b = 5 : 4 → a = (5/4) * 36 = 45. From b : c = 9 : 7 → c = (7/9) * 36 = 28. Hence, a : b : c = 45 : 36 : 28.

Verification / Alternative check: Check the original equations: 4a = 4*45 = 180; 5b = 5*36 = 180 (ok). Also 7b = 7*36 = 252; 9c = 9*28 = 252 (ok).

Why Other Options Are Wrong:

• Other permutations do not satisfy both equations simultaneously.
• 44 : 33 : 28 is not consistent with 4a = 5b.

Common Pitfalls: Not equalizing b properly or mixing the roles of b and c when translating ratios. Always verify by substituting back into the original equations.

Final Answer: 45 : 36 : 28