Working on alternate days: Alen can do a job in 21 days and Border in 42 days working alone. If they work on alternate days starting with Alen (day 1), in how many days will the work be completed?

Introduction / Context:
Alternate-day problems are solved by computing the work done in a 2-day cycle and then scaling. Starting worker matters if the total is not a multiple of the cycle output, but this pair divides nicely.

Given Data / Assumptions:

• Alen alone: 21 days ⇒ rate_A = 1/21.
• Border alone: 42 days ⇒ rate_B = 1/42.
• They alternate days starting with Alen.

Concept / Approach:
Compute 2-day work = 1/21 + 1/42 = 3/42 = 1/14 of the job. One full job needs 14 such 2-day cycles if it divides exactly.

Step-by-Step Solution:
2-day output = 1/21 + 1/42 = 1/14To complete 1 job: need 14 cycles × 2 days = 28 days

Verification / Alternative check:
After 28 days, the total done is 14 * (1/14) = 1 job. No fractional day is needed because the cycle output divides the job exactly.

Why Other Options Are Wrong:

• 14: would require 1/14 per day, but they achieve 1/14 in two days, not one.
• 42 and 35: are longer than the exact computed 28 days.

Common Pitfalls:

• Assuming linear averaging of times instead of cycle-based aggregation.

Final Answer:
28