Percent-to-ratio conversion between two numbers: One-half of a certain first number equals 65% of a second number. Determine the ratio of the first number to the second number (first : second), showing the percentage-to-decimal transition.

Introduction / Context:
Converting a relation containing a fraction and a percentage into a clean ratio is a standard proportional skill. Translate percent to decimal and then isolate the ratio between the two numbers to simplest integers.

Given Data / Assumptions:

• (1/2)*First = 65% of Second = 0.65*Second.

Concept / Approach:
Start from (1/2)A = 0.65B. Multiply both sides to isolate A/B. Express as a fraction and then reduce to the smallest integer ratio.

Step-by-Step Solution:

Verification / Alternative check:
Let B = 10 ⇒ A = 13. Then (1/2)A = 6.5 and 65% of B = 6.5, consistent.

Why Other Options Are Wrong:
10 : 13, 8 : 13, 13 : 8, 5 : 13 invert or distort the 1.30 multiplier connecting A and B.

Common Pitfalls:
Treating 65% as 0.65 of the first number or mixing the sides; ensure the fraction applies to the correct variable.

Final Answer:
13 : 10