Contract sharing with a helper: Three persons take a job for $1200. Their individual times are 8 days, 12 days, and 16 days. With the help of a fourth person, they finish the entire job in 3 days. Find the fair payment (share) for the fourth person.

Introduction / Context:
In mixed-time work problems, payments are proportional to the actual work done. Given individual capacities (days to complete alone), we can convert to daily work rates, add the fourth person’s unknown rate, and use the total time to finish to determine the helper’s contribution and hence his share of the money.

Given Data / Assumptions:

• Total contract amount = $1200.
• Worker A alone: 8 days ⇒ rate = 1/8 job/day.
• Worker B alone: 12 days ⇒ rate = 1/12 job/day.
• Worker C alone: 16 days ⇒ rate = 1/16 job/day.
• All four together finish in 3 days.
• Shares are proportional to individual work done.

Concept / Approach:
Let the fourth person’s daily rate be x. Total daily rate = 1/8 + 1/12 + 1/16 + x. Over 3 days, total work equals 1 job. Solve for x. Then compute the fourth person’s fraction of the job completed and multiply by $1200 to get his share.

Step-by-Step Solution:

Verification / Alternative check:
The other three together provide 13/48 per day; adding 1/16 gives 16/48 = 1/3 per day; in 3 days, they complete exactly 1 job.

Why Other Options Are Wrong:
$200 and $180 undervalue the helper’s exact fraction; $250 overstates it; $240 is not aligned with 3/16 of $1200.

Common Pitfalls:
Splitting money by days participated rather than actual work fractions, or mis-adding the reciprocals.

Final Answer:
$225