The time taken by 4 men to complete a certain job is double the time taken by 5 children to complete the same job. Each man works at twice the rate of a woman, and each child can finish the entire job alone in 20 days. If 12 men, 10 children, and 8 women work together, in how many days will they complete the job?

Introduction / Context:

This question combines multiple layers of time and work reasoning. It connects the relative efficiencies of men, women, and children using conditions on completion times and then asks for the time needed when all three categories work together. The problem tests comfort with work rates, ratios, and translating relationships such as "twice as fast" into mathematical form.

Given Data / Assumptions:

• Four men take twice as long as five children to complete the same job.
• Each man is twice as fast as each woman.
• Each child can complete the job alone in 20 days.
• We must find the time taken when 12 men, 10 children, and 8 women work together.
• Work is uniform and the job size is taken as one unit.

Concept / Approach:

We first convert the given times into work rates for children, then use the comparative time relation between men and children to find the rate of a man. Next we use the information that a man is twice as fast as a woman to find the woman's rate. Finally we add the work rates of all 12 men, 10 children, and 8 women to get a combined rate, and then take the reciprocal to find the total time required to finish the job together.

Step-by-Step Solution:

Verification / Alternative Check:

If all workers complete exactly one unit of work per day, by definition the job is done in one day. The intermediate relationships also check out: four men at 1/32 each give 1/8 per day and hence 8 days for the full job, double the 4 days taken by five children. The man to woman efficiency ratio is consistent with the derived rates of 1/32 and 1/64 respectively. Thus all constraints are satisfied.

Why Other Options Are Wrong:

Options such as 2, 3, or 4 days would correspond to combined daily work rates of 1/2, 1/3, or 1/4 of the job respectively, which contradict the calculated sum of individual rates. Since we have shown that the total combined rate is exactly 1 unit per day, any answer other than 1 day is inconsistent with the given relationships.

Common Pitfalls:

Candidates sometimes misinterpret the statement that four men take double the time needed by five children, mistakenly equating the number of men with the number of children instead of time. Another frequent error is assuming that a man does double the work of a child, which is not stated. Only the relation between men and women is given. Correctly following each given condition is critical for reaching the right formulae and final rate.

Final Answer:

Working together, the group of 12 men, 10 children, and 8 women will finish the job in 1 Day.