Chain ratios: If a : b = 3 : 5 and b : c = 4 : 7, determine a : c.

Introduction / Context:
Combining two ratios with a common middle term (here b) requires equalizing that middle term. This is the standard method to derive a chain ratio such as a : c from a : b and b : c.

Given Data / Assumptions:
a : b = 3 : 5 and b : c = 4 : 7.

Concept / Approach:
Adjust the ratios so that the b term is the same in both. Then multiply across to get a consistent a : b : c, and finally extract a : c.

Step-by-Step Solution:
From a : b = 3 : 5, take b = 5k → a = 3k. From b : c = 4 : 7, take b = 4m → c = 7m. Equalize b: 5k = 4m. Choose k = 4 and m = 5 → b = 20. Then a = 3k = 12 and c = 7m = 35. Therefore, a : c = 12 : 35.

Verification / Alternative check:
Build the full triple ratio: a : b : c = 12 : 20 : 35. Check a : b = 12 : 20 = 3 : 5 and b : c = 20 : 35 = 4 : 7, confirming consistency.

Why Other Options Are Wrong:

• 11 : 35 and 35 : 11 are not derived by equalizing b.
• 35 : 12 flips order and values.
• 3 : 7 would be correct only if a : b and b : c were compatible with that direct relation, which they are not without scaling.

Common Pitfalls:
Failing to equalize the common term or incorrectly cross-multiplying. Always align the shared term before concluding the end ratios.

Final Answer:
12 : 35