Two men with a helper (money division by work): The first man alone can finish a work in 7 days and the second man alone in 8 days. Together with a boy they finish it in 3 days. If the total amount is to be divided proportional to work done, which split (in dollars) is correct?

Introduction / Context:
When a lump sum is split by contribution, each person’s share equals their fraction of the total work. With two men of known solo times and a boy helping so the job finishes in 3 days, we derive each one’s share fraction and compare with the given dollar splits to identify the correct one.

Given Data / Assumptions:

• Rate(man1) = 1/7 per day
• Rate(man2) = 1/8 per day
• Total rate with boy = 1/3 per day
• Division is proportional to work done in 3 days

Concept / Approach:
Boy’s rate = total rate − (man1 + man2) rate. Each share fraction equals 3 * (individual rate), since they work for 3 days, and these fractions sum to 1. Reduce to a clean integer ratio and see which option matches.

Step-by-Step Solution:
Rate(man1) + Rate(man2) = 1/7 + 1/8 = (8 + 7)/56 = 15/56Total rate = 1/3 = 56/168; (man1 + man2) = 45/168Rate(boy) = 56/168 − 45/168 = 11/168Share fractions over 3 days: man1 = 3*(1/7) = 3/7; man2 = 3*(1/8) = 3/8; boy = 3*(11/168) = 33/168Convert to integer ratio: multiply by 168 ⇒ 72 : 63 : 33 ⇒ simplify to 24 : 21 : 11Check option splits for this 24:21:11 ratio → $600:$525:$275 fits exactly

Verification / Alternative check:
Any total consistent with the 24:21:11 ratio yields the same proportional split; the provided option matches this ratio precisely.

Why Other Options Are Wrong:
Other splits reduce to different ratios (e.g., 6:5:3 or 12:11:5), not 24:21:11.

Common Pitfalls:
Forgetting to multiply rates by the common duration (3 days); arithmetic slips when reducing fractions; comparing shares by amounts instead of checking ratios.

Final Answer:
$600, $525, $275