Relative subtraction and resulting multiple: From each of two numbers, half the smaller is subtracted. After this, the larger becomes exactly 4 times the smaller. Determine the ratio of the original numbers (larger : smaller).

Introduction / Context:
This problem encodes a before-and-after transformation. By writing the post-subtraction values explicitly and applying the given multiplicative relation, you can form a linear equation connecting the original numbers, then express the ratio.

Given Data / Assumptions:

• Original larger = L; original smaller = S (L > S > 0).
• Subtract (1/2)S from each.
• Resulting larger = L − (1/2)S; resulting smaller = S − (1/2)S = (1/2)S.
• After subtraction: resulting larger = 4 * resulting smaller.

Concept / Approach:
Set L − (1/2)S = 4 * (1/2)S and solve for L in terms of S. Then read the ratio L : S.

Step-by-Step Solution:

Verification / Alternative check:
Example: let S = 2; then L = 5. After subtraction: new larger = 5 − 1 = 4; new smaller = 2 − 1 = 1; indeed 4 = 4 * 1.

Why Other Options Are Wrong:
1 : 4 and 4 : 5 invert or misorder; 4 : 1 ignores the half-subtraction; 3 : 2 does not satisfy the post-subtraction multiple condition.

Common Pitfalls:
Subtracting half of the larger instead of half of the smaller from both numbers, or misidentifying which becomes 4 times which after the operation.

Final Answer:
5 : 2