A and B take a $4500 contract; A alone 8 days, B alone 12 days: With C's help they finish in 4 days. Assuming payment is proportional to work done, what is C's share of the money?

Introduction / Context:
In joint work, individual daily work rates add. Once the combined completion time is known, each worker’s portion of the total work is rate * time. Payment is then allocated in that same proportion.

Given Data / Assumptions:

• A alone finishes in 8 days ⇒ rate(A) = 1/8 work/day.
• B alone finishes in 12 days ⇒ rate(B) = 1/12 work/day.
• A + B + C finish in 4 days ⇒ total rate = 1/4 work/day.
• Total payment = $4500.

Concept / Approach:
Find C's daily rate by subtraction from the combined rate. Then compute C's share of total work in 4 days and multiply by the contract amount.

Step-by-Step Solution:
rate(A) = 1/8, rate(B) = 1/12.rate(A) + rate(B) = 1/8 + 1/12 = 3/24 + 2/24 = 5/24.rate(A+B+C) = 1/4 ⇒ rate(C) = 1/4 − 5/24 = (6 − 5)/24 = 1/24.C's share of work over 4 days = 4 * (1/24) = 1/6 of the job.C's share of money = (1/6) * 4500 = $750.

Verification / Alternative check:
A’s share of work in 4 days = 4*(1/8) = 1/2; B’s = 4*(1/12) = 1/3; C’s = 1/6. These add to 1, validating the rates. The payouts in the same ratio sum to $4500.

Why Other Options Are Wrong:

• $2250 and $1500 are too large for C's small rate.
• $375 is half of the correct $750, inconsistent with rate(C) = 1/24.

Common Pitfalls:

• Using 4 days as a divisor for money directly without computing each worker’s share of the job.

Final Answer:
$750