Time relations relative to the team: A alone takes 25 days more than (A + B) together, and B alone takes 49 days more than (A + B) together. How many days do A and B together take to finish the work?

Introduction / Context: This problem defines solo times in terms of the team time. Let t be the time taken by A + B together; then A alone is t + 25 and B alone is t + 49. Setting the sum of solo rates equal to the team rate yields a solvable equation for t.

Given Data / Assumptions:

• (A + B) time = t days.
• A alone time = t + 25 days.
• B alone time = t + 49 days.
• Work size = 1 unit.

Concept / Approach: Rates add: 1/(t + 25) + 1/(t + 49) = 1/t. Clear denominators to form a quadratic in t, then solve for the positive root to find the team time directly.

Step-by-Step Solution:

Verification / Alternative check: A alone = 60 days, B alone = 84 days. Check: 1/60 + 1/84 = (7 + 5)/420 = 12/420 = 1/35, matching the team time of 35 days.

Why Other Options Are Wrong: 25, 15, or 45 days do not satisfy the relation between solo and team times defined in the equation; only 35 days does.

Common Pitfalls: Adding times instead of rates or making algebraic errors while cross-multiplying. Keep denominators intact until the final quadratic emerges.

Final Answer: 35 days