Number system — two-number word problem with proportions and a linear condition: 1/5 of the first number equals 5/8 of the second number. If 35 is added to the first number, the result becomes 4 times the second number. Find the exact value of the second number.

Introduction / Context:
This problem combines a ratio (proportional) relationship between two unknown integers with a linear shift condition. Such questions commonly appear in aptitude tests to assess fluency with translating words into equations and solving a small system consistently.

Given Data / Assumptions:

• Let the first number be x and the second number be y.
• Condition 1: (1/5) * x = (5/8) * y.
• Condition 2: x + 35 = 4 * y.
• All quantities are real numbers; we anticipate integer answers from tidy ratios.

Concept / Approach:
Convert each sentence into an equation and solve the two equations in two unknowns. From the proportional statement we isolate x in terms of y, then substitute into the linear shift condition. This avoids handling two variables simultaneously in multiple places and reduces arithmetic errors.

Step-by-Step Solution:
From (1/5) * x = (5/8) * y, multiply both sides by 5: x = (25/8) * y.Use the second condition: x + 35 = 4y.Substitute x: (25/8) * y + 35 = 4y.Move terms: 35 = 4y − (25/8) * y = (32/8 − 25/8) * y = (7/8) * y.Solve for y: y = 35 * 8 / 7 = 40.

Verification / Alternative check:
Compute x from x = (25/8) * y = (25/8) * 40 = 125. Check the second condition: 125 + 35 = 160 and 4 * y = 4 * 40 = 160, which matches perfectly.

Why Other Options Are Wrong:
125: That is the first number x, not the second number y.70 or 25: Substituting either breaks the proportional or the linear condition; they do not satisfy both simultaneously.

Common Pitfalls:
Mixing up which variable is the first or second number; forgetting to multiply both sides evenly when clearing denominators; skipping simplification of fractional coefficients leading to arithmetic slips.

Final Answer:
40