How to Solve Work-and-Wages Sharing with Partial Participation
Solve work-and-wages sharing aptitude problems with partial participation using contribution-value ratios, with a worked example.
Expected Interview Answer
When workers join or leave partway through a job, or contribute unequal days, wages are shared strictly by the actual fraction of total work each person completed — computed as their one-day-work rate multiplied by the exact number of days they worked, then normalized against the sum of everyone’s contribution.
Start from each worker’s one-day-work rate (1 divided by their solo completion time). Multiply each worker’s rate by the exact number of days they were actually present on the job — not the total project duration if they left early or joined late. Sum all these individual contributions to confirm they total 1 (the whole job); if they do not, recheck the days or rates. Wages are then split in the exact ratio of these individual contribution values, which is not the same as splitting by days present alone, since a more efficient worker present for fewer days can still deserve a larger share.
- Handles unequal presence duration alongside unequal efficiency correctly
- Contribution values double as a built-in check (they must sum to 1)
- Prevents the common error of splitting wages by days present alone
AI Mentor Explanation
If one bowler bowls the first 8 overs of a 20-over innings at a rate that would take 25 overs alone to bowl the full spell equivalent, and a second bowler covers the remaining 12 overs at a rate that would take 15 overs alone, their contributions to a shared man-of-the-match bonus are not split by overs bowled (8:12) but by actual work done: (8/25) versus (12/15) — computed from each bowler's efficiency times their actual overs, then normalized. Wage-sharing with partial participation always uses this contribution-value method, never raw time-present alone.
Worked example (partial participation)
A contribution
- 8 × 1/20 = 0.4
B contribution
- 18 × 1/30 = 0.6
Wage split (ratio 2:3)
- A = 400, B = 600
Step-by-Step Explanation
Step 1
Find each worker's daily rate
One-day-work = 1 ÷ (days to complete the job alone).
Step 2
Multiply by actual days present
Contribution = daily rate × exact number of days that worker actually worked.
Step 3
Verify contributions sum to 1
All individual contributions must add to the whole job (1); mismatches signal an error in days or rates.
Step 4
Split payment by the contribution ratio
Divide the wage in the exact ratio of each worker's contribution value, not by days present alone.
What Interviewer Expects
- Correct daily-rate computation for each worker
- Correct multiplication by actual days present, not total project duration
- Verification that contributions sum to the whole job
- Wage split strictly by contribution ratio, distinguishing it from a days-present split
Common Mistakes
- Splitting wages by days present instead of actual work contributed
- Using total project duration instead of each worker's actual days present
- Forgetting to verify that contributions sum to 1 as a sanity check
- Confusing a worker's daily rate with their total contribution value
Best Answer (HR Friendly)
“When people join or leave a job partway through, I do not split pay by how many days each one was around — I compute what fraction of the actual job each person completed. That is their daily rate times the exact days they worked, and all those fractions together should add up to the whole job as a sanity check. Then I split the payment in that same contribution ratio, which can favor someone who worked fewer days but was more efficient.”
Follow-up Questions
- How would this change if a third worker joins midway through the remaining work?
- How do you verify your contribution values are correct before splitting wages?
- What if the total work does not sum to exactly 1 — how do you find the missing days?
- How does this generalize to a case where a worker's efficiency changes partway through?
MCQ Practice
1. A can finish a job alone in 15 days, B in 10 days. A works alone for 5 days, then B joins and they finish together. A's contribution to the total work is?
A's contribution = 5 days × 1/15 = 5/15 = 1/3, matching option value 5/15 which simplifies to 1/3.
2. A takes 12 days alone, B takes 24 days alone. A works 4 days, then B finishes the rest alone. A payment of 900 for the whole job is split in what ratio (A:B)?
A contributes 4/12 = 1/3 of the job; B contributes the remaining 2/3. Ratio A:B = 1/3 : 2/3 = 1:2.
3. Why must individual contributions in a partial-participation problem sum to 1?
The contributions represent fractions of one complete job; if they do not sum to 1, the job was not fully completed as described, signaling an error.
Flash Cards
How to compute a worker's contribution? — Daily rate (1 / days alone) multiplied by the exact number of days actually worked.
Sanity check for partial-participation problems? — All workers' contributions must sum to exactly 1 (the whole job).
How are wages split? — In the exact ratio of each worker's contribution value, not by days present.
Why can a worker present fewer days earn more? — Because higher efficiency can produce a larger contribution value despite fewer days present.