Duplicate ratio: What is the duplicate ratio (i.e., the squared ratio) of 2 : 7?

Introduction / Context:
Classical ratio terminology: the duplicate ratio of a : b is the ratio of their squares, a^2 : b^2. Similarly, sub-duplicate refers to square roots, and triplicate refers to cubes. This problem asks for the duplicate ratio of 2 : 7.

Given Data / Assumptions:

• Base ratio = 2 : 7.
• Duplicate ratio means square each term.

Concept / Approach:
If the ratio is a : b, then the duplicate ratio is a^2 : b^2. No simplification is necessary if the squared terms are already co-prime or in simplest form.

Step-by-Step Solution:
Start with 2 : 7. Square both terms: 2^2 : 7^2 = 4 : 49. The pair (4, 49) shares no common factor other than 1, so it is already in simplest terms.

Verification / Alternative check:
Numerically, 2/7 ≈ 0.2857. The duplicate ratio is (2/7)^2 ≈ 0.08163. As a ratio of integers with the same value, 4/49 ≈ 0.08163, confirming correctness.

Why Other Options Are Wrong:

• 49 : 4 is the inverse duplicate ratio.
• 4 : 14 mixes squaring with doubling.
• 8 : 343 is the triplicate ratio (2^3 : 7^3).
• 14 : 49 mixes multiplication with squaring.

Common Pitfalls:
Confusing duplicate with triplicate ratio, or accidentally inverting the order. Remember: duplicate = square each term, keep the original order.

Final Answer:
4 : 49