Three men, four women and six children can complete a work in seven days. A woman does double the work a man does and a-child does half the work a man does. How many women alone can complete the work in 7 days?

According to the question: 

Total work = (3M + 4W + 6C) * 7

Now, 

A woman does double the work a man does, So, the ratio of efficiency of a man to that of a woman = 1: 2

M: W = 1: 2

2M = W .......... (1)

A child does half the work a man does. So, the ratio of efficiency of a man to that of child = 2: 1

M: C = 2: 1

M = 2C ....... (2)

From (1) and (2):

4C = W ....... (3)

Let the required number of women = 'x'

Now, 

(3M + 4W + 6C) * 7 = (xW) * 7

From (1) and (3):

1.5W + 4W + 1.5W = xW

7W = xW

x = 7