If 2A = 3B and 4B = 5C, find the ratio A : C.

Introduction / Context:Linked proportional equations can be combined to relate two variables that do not appear together initially. Express A and C in terms of B (or a common parameter) and then form their ratio.

Given Data / Assumptions:

• 2A = 3B ⇒ A = (3/2)B
• 4B = 5C ⇒ C = (4/5)B
• All variables are positive.

Concept / Approach:Write A and C each as a multiple of B. Then A : C = [(3/2)B] : [(4/5)B] = (3/2) : (4/5) = (3/2) * (5/4) = 15/8.

Step-by-Step Solution:A = (3/2)BC = (4/5)BA : C = (3/2) : (4/5) = (3/2)*(5/4) = 15/8

Verification / Alternative check:Choose B = 40 (LCM approach). Then A = (3/2)*40 = 60; C = (4/5)*40 = 32; A : C = 60 : 32 = 15 : 8.

Why Other Options Are Wrong:3 : 4 and 4 : 3 invert or mismatch; 8 : 15 is the reciprocal; 5 : 6 does not satisfy the derived relation.

Common Pitfalls:Dividing instead of multiplying by reciprocals when forming the ratio; losing the common factor B too early.

Final Answer:15 : 8