Different efficiencies, combined time known: Two friends A and B working together finish a job in 26 days. Their efficiencies (work rates) are in the ratio 8 : 5. If B works alone, how many days will he take to complete the job?

Introduction / Context:
When two workers have rates in a given ratio and their combined time is known, we can express their individual rates as proportional parts of the combined rate. From there, the solo completion time for either worker follows by taking the reciprocal of the individual rate.

Given Data / Assumptions:

• Together time = 26 days ⇒ combined rate = 1/26 work/day.
• Rate ratio A : B = 8 : 5.
• Let A’s rate = 8k, B’s rate = 5k (work/day).

Concept / Approach:
Since 8k + 5k = 13k equals the combined rate 1/26, we can solve for k. Then B’s rate is 5k, and B’s required time is 1 divided by that rate. Keep units consistent as work/day.

Step-by-Step Solution:

Verification / Alternative check:
A’s rate would be 8/338; combined with B’s 5/338 gives 13/338 = 1/26, matching the given teamwork time, so the proportioning is correct.

Why Other Options Are Wrong:
Values like 105 1/4 or 112 3/5 arise from arithmetic slips or wrong reciprocals; 214 2/3 and 52 days are inconsistent with the proportional split of the 1/26 rate.

Common Pitfalls:
Conflating time ratios with rate ratios. Remember, rates add; times do not. Use the combined rate to determine the constant of proportionality.

Final Answer:
67 3/5 days