Two men and a boy complete a job: Two men undertake a piece of work for $600. Individually, they can finish it in 6 days and 8 days respectively. With the help of a boy, they complete the work in 3 days. What is the boy’s share (in dollars)?

Introduction / Context:
We distribute a fixed amount ($600) according to each participant’s contribution to the work. Two men of known solo times and a boy together finish in 3 days, letting us compute the boy’s rate and thus his monetary share.

Given Data / Assumptions:

• Rate(man1) = 1/6 per day
• Rate(man2) = 1/8 per day
• Total rate with boy = 1/3 per day
• Total amount to share = $600

Concept / Approach:
Find the boy’s daily rate by subtracting the two men’s combined rate from the total. The share fraction for each person equals their rate divided by the total rate, or equivalently rate * 3 days since they all worked 3 days. Multiply the boy’s fraction by $600 to get his share.

Step-by-Step Solution:
Rate(man1) + Rate(man2) = 1/6 + 1/8 = (4 + 3)/24 = 7/24Total rate = 1/3 = 8/24Boy’s rate = 8/24 − 7/24 = 1/24Boy’s share fraction = (1/24) / (1/3) = (1/24) * 3 = 1/8Boy’s share = 1/8 * $600 = $75

Verification / Alternative check:
Men’s fractions: (1/6)/(1/3) = 1/2 and (1/8)/(1/3) = 3/8. Together with 1/8 sum to 1; shares are $300, $225, and $75, totaling $600.

Why Other Options Are Wrong:
$300 and $225 belong to the two men; $100 would overpay the boy versus his 1/8 contribution.

Common Pitfalls:
Allocating by days rather than rates; forgetting to normalize by the total combined rate; arithmetic slips with fractions.

Final Answer:
$75