Two men or five women or seven boys can finish a work in 469 days. In how many days will seven men, five women, and two boys together finish the same work?

Introduction / Context:
We are given interchangeable work groups. From the common completion time, we deduce each individual’s daily work rate and then sum mixed rates to get the combined completion time.

Given Data / Assumptions:

• 2 men finish in 469 days.
• 5 women finish in 469 days.
• 7 boys finish in 469 days.
• Find days for 7 men + 5 women + 2 boys.

Concept / Approach:
If a group finishes in D days, daily group rate = 1/D of the work. Hence one man’s daily rate = (1/469)/2, etc. After finding each single rate, sum the mixed team’s daily rate and invert to get days.

Step-by-Step Solution:
Let total work = 1. 2 men rate = 1/469 ⇒ 1 man = 1/938 per day. 5 women rate = 1/469 ⇒ 1 woman = 1/2345 per day. 7 boys rate = 1/469 ⇒ 1 boy = 1/3283 per day. Mixed team rate = 7*(1/938) + 5*(1/2345) + 2*(1/3283). Compute exactly: this sum simplifies to 1/98 per day (can be checked by common-denominator arithmetic). Required days = 1 / (1/98) = 98 days.

Verification / Alternative check:
Using a calculator or fraction arithmetic confirms the sum equals 1/98. Hence the time is 98 days.

Why Other Options Are Wrong:
134, 106, 100, 96 days do not match the combined rate computed from the equivalences provided.

Common Pitfalls:
Assuming proportionality between numbers of workers and days without establishing consistent single-worker rates; forgetting to invert summed rate to get time.

Final Answer:
98 days