This is merely a subtype of Time n Work problems. and can be solved using our good ol’ STD table Method.

# Case 1: Women finish entire job

4 men and 6 women can complete a work in 8 days, while 3 men and 7 women can complete it in 10 days. In how many days will 10 women complete it?

Suppose 1 man can pour 1 bucketful of water in the tank in 1 minute.
If 6 men work together, they pour 6 buckets in a tank in 1 minute.
In short, you multiply the speed with number of person.

Our usual table

 1 Man 1 Woman 4 men & 6 women 3 men & 7 women Speed m w 4m+6w 3m+7w Time 8 10 distance

Distance covered in each column is same.
So compare last two columns
Distance = distance
Speed x time = speed x time
(4m+6w)*8=(3m+7w)*10

Solve this equation and you get m=11w
Replace this value of w in last column

 1 Man 1 Woman 4 men & 6 women 3 men & 7 women Speed m w 4m+6w 3(11w)+7w Time 8 10 distance

3(11w)+7w
=33w+7w
=40w

 1 Man 1 Woman 4 men & 6 women 3 men & 7 women Speed m w 4m+6w 40w Time 8 10 distance 40w x 10= 400w

So the total work is
Speed x time = distance
40w x 10= 400w

If 10 women work together, they’ve to cover 400w kms. Make a new column, run STD formula

 1 Man 1 Woman 4 men & 6 women 3 men & 7 women 10 women Speed m w 4m+6w 40w 10w Time 8 10 ?? distance 40w x 10= 400w 400w

10w x time =400w
Time =400w/ 10w=40days.

Answer: If 10 women work together, it’ll take 40 days to finish the job.

# Case 2: Women to finish remaining job

12 men can complete a piece of work in 4 days, while 15 women can complete the same work in 4 days. 6 men start working on the job and after working for 2 days, all of them stopped working. How many women should be put on the job to complete the remaining work, if it is to be completed in 3 days?

 12 Man 15 Woman Speed 12m 15w Time 4 4 distance

Distance covered in both column is same so compare them

Speed x time = speed x time
12m x 4= 15w x 4
4m=5w
M=(5/4)w

The question is asking about number of women in the end, so better convert everything in terms of women.
Given:  6 men start working on the job and after working for 2 days, all of them stopped working
Make a new column.

 12 Man 15 Woman 6men Speed 12m 15w 6x (5/4)w Time 4 4 2 distance 60w 6x (5/4)w x2=15w

As you can see, I’ve applied STD formula in last two columns to find distance simultaneously.
Remaining work=60w minus 15w=45w
Required: we want to finish this work in 3 days.
Asked: How many women should be put on the job to complete the remaining work, if it is to be completed in 3 days?

Make a new column. Suppose we need “F” number of women so speed = f x w = Fw.

 12 Man 15 Woman 6men Find women Speed 12m 15w 6x (5/4)w Fw Time 4 4 2 3 Distance 60w 15w 45w

Time limit is given to us: complete remaining work in three days. So t=3 for last column.
Run STD on last column
Fw x 3 =45w
F=45w /3w
F=15

Important: Remaining work is to be completed in 3 days. So t=3
If they had asked “total work is to be completed in 3 days”, we’ll need to take t=1, because those men already worked for 2 days to only 1 day left to complete the job (2+1=3 days)

# Case: Child labour

Twelve children take sixteen days to complete a work which can be completed by eight adults in twelve days. Sixteen adults started working and after three days ten adults left and four children joined them. How many days will they take to complete the remaining work?

 12 kids 8 men Speed 12k 8m Time 16 12 Distance

Since work done in both columns is same
Speed x time = speed x time
12k x 16= 8m x 12
K=(1/2)m

Concentrate on middle column (8 men) run STD table on it. You get distance = 8m x 12 =96m
So total distance to be covered is 48m

Given: Sixteen adults started working and after three days ….
Means 16 men worked for 3 days.

 12 kids 8 men 16 men Speed 12k 8m 16m Time 16 12 3 Distance 8m x 12=96m 16m x 3=48m

We’ve run STD on last column and got that 16 men covered 48m kms.
So remaining work is 96m minus 48m=48m

Given: after three days ten adults left and four children joined them
So now men left =16 minus 10 =6 men and 4 kids joins them so speed is 6m+4k
Make a new column

 12 kids 8 men 16 men Men & kids Speed 12k 8m 16m 6m+4k Time 16 12 3 ?? Distance 96m 48m 48m

Run STD on last column
(6m+4k) x time =48m
But We already calculated that K=(1/2)m
Apply it in above equation{6m+4(1/2)m}  x time =48m
{6m+2m} x time = 48m
8m x time = 48m
Time =48m /8m
Time=6 days.