If a : b = 2 : 3 and 2 : a = 1 : 2, find the value of b.

Introduction / Context:
Two ratio statements allow solving for a and then b concretely. The second ratio fixes a numerically, and the first then gives b from a standard proportion.

Given Data / Assumptions:

• a : b = 2 : 3
• 2 : a = 1 : 2

Concept / Approach:
From 2 : a = 1 : 2, write 2/a = 1/2 ⇒ a = 4. Substitute into a : b = 2 : 3 to find b.

Step-by-Step Solution:
2/a = 1/2 ⇒ a = 4a : b = 2 : 3 ⇒ 4 : b = 2 : 3Cross-multiply: 4*3 = 2*b ⇒ b = 12/2 = 6

Verification / Alternative check:
Check both ratios with a = 4, b = 6: a : b = 4 : 6 = 2 : 3; 2 : a = 2 : 4 = 1 : 2. Both satisfied.

Why Other Options Are Wrong:
4, 1/3, 3/2, 8 contradict one or both given ratios when substituted.

Common Pitfalls:
Interpreting 2 : a = 1 : 2 as 2/a = 2/1 (reversing); skipping the final consistency check.

Final Answer:
6