Percent-to-ratio conversion: Thirty-five percent of one number equals twice seventy-five percent of another number. Find the ratio of the first number to the second number.

Introduction / Context:
Translating percentage statements into algebraic equations enables quick ratio determination. This question asks you to equate percent portions of two unknown numbers and express the result as a clean integer ratio.

Given Data / Assumptions:

• Let the first number be x and the second be y.
• 35% of x equals twice 75% of y.

Concept / Approach:
Convert percentages to decimals and set up the equation. Then isolate x/y to get a ratio. Multiply numerator and denominator to clear decimals and express the ratio with integers in lowest terms.

Step-by-Step Solution:
0.35x = 2 * 0.75y ⇒ 0.35x = 1.5y.x / y = 1.5 / 0.35.1.5 / 0.35 = 150 / 35 = 30 / 7.Therefore, the required ratio is 30 : 7.

Verification / Alternative check:
Pick y = 7 (for convenience). Then x = 30 satisfies 0.35 * 30 = 10.5 and 2 * 0.75 * 7 = 10.5, verifying the equality precisely.

Why Other Options Are Wrong:

• 35 : 6, 23 : 7, and 32 : 9 do not satisfy 0.35x = 1.5y when tested.

Common Pitfalls:

• Mishandling percent-to-decimal conversion (e.g., mixing 35 with 0.35).
• Forgetting to simplify the final ratio to lowest integer terms.

Final Answer:
30 : 7