Chaining three ratios to find A : D: If A : B = 2 : 3, B : C = 5 : 7, and C : D = 3 : 10, determine A : D in simplest form.

Introduction / Context:
This problem requires careful linking of three ratios through the repeated variables B and C to arrive at the ratio between A and D. It highlights consistent scaling across chained proportional relations.

Given Data / Assumptions:

• A : B = 2 : 3.
• B : C = 5 : 7.
• C : D = 3 : 10.

Concept / Approach:
Equate the middle terms successively. Express each variable in terms of a single base quantity (e.g., x) and simplify the final ratio A : D.

Step-by-Step Solution:
Let A = 2x, B = 3x.From B : C = 5 : 7 ⇒ if B = 3x = 5y, then y = 3x/5 and C = 7y = 21x/5.From C : D = 3 : 10 ⇒ if C = 21x/5 = 3z, then z = 7x/5 and D = 10z = 14x.Therefore A : D = 2x : 14x = 1 : 7.

Verification / Alternative check:
Normalize to x = 5 for easy integers: A = 10, D = 70 ⇒ A : D = 1 : 7 as obtained.

Why Other Options Are Wrong:

• 2 : 7, 1 : 5, 5 : 1 are not consistent with the chained scaling.

Common Pitfalls:

• Losing track of scaling constants when switching between the three ratios.
• Failing to simplify at the end.

Final Answer:
1 : 7