Three men, four women and six children can together complete a piece of work in 9 days. Each woman does double the work that one man does, and each child does half the work that one man does. How many women working alone can complete the same work in 9 days?

Introduction / Context:
This time and work question checks your understanding of equivalent work rates and how to convert a mixed group of men, women and children into a single type of worker. Such problems are very common in competitive exams, and the key idea is to express the work done by every category of worker in terms of one common base unit so that comparisons and calculations become simple.

Given Data / Assumptions:

• Three men, four women and six children can complete the work in 9 days.
• One woman does double the work that one man does.
• One child does half the work that one man does.
• We assume constant efficiency for all workers throughout the work period.
• We need the number of women who can alone complete the same work in 9 days.

Concept / Approach:
The standard approach is to choose one worker type as the base unit of efficiency. Here, we take one man's daily work as 1 unit. Then we convert the work done by a woman and a child into equivalent man units. Once we know the total work in man units, we convert it into woman units and compute how many women working together for 9 days can finish the job.

Step-by-Step Solution:
Let the daily work done by one man = 1 unit. Then daily work done by one woman = 2 units (given as double a man). Daily work done by one child = 0.5 units (half the work of a man). Group daily work = 3 men + 4 women + 6 children. = 3 * 1 + 4 * 2 + 6 * 0.5 = 3 + 8 + 3 = 14 units per day. Total work in units = 14 * 9 = 126 units. Let the number of women needed = n. Daily work of n women = n * 2 units. In 9 days, work done by n women = n * 2 * 9 = 18n units. This must equal total work: 18n = 126. So, n = 126 / 18 = 7.

Verification / Alternative check:
We can check quickly: Seven women do 7 * 2 = 14 units per day, exactly the same daily work as the original group of 3 men, 4 women and 6 children. Since both do 14 units per day, and the original group completed the work in 9 days, seven women will also complete the work in 9 days. This confirms that our calculation is consistent and correct.

Why Other Options Are Wrong:
Each of 5, 6 and 8 women working alone would produce a different daily output from 14 units per day. For example, 6 women would do 12 units per day and take more than 9 days, while 8 women would do 16 units per day and finish in fewer than 9 days. Hence these options do not match the given condition of finishing in 9 days.

Common Pitfalls:
A common mistake is to forget that a woman and a child have different efficiencies from a man and to simply add the headcount. Another frequent error is to ignore the requirement that the work must still be completed in exactly 9 days, which is crucial for forming the equation. Carefully converting all workers into equivalent units avoids these errors.

Final Answer:
The number of women who can complete the work in 9 days is 7.