R.k91 wrote:If 2 men and 5 women do apiece of work in 8 days and 2 men and 7 children do the same piece of work in 12 days. show that 10 women can do as much work in 9 days as 31 children.

Hi, there. I'm happy to help with this.

This is a work question, a frequent word problem on the GMAT Quantitative.

The BIG idea with work questions: you can't add or subtract times to work or pieces of work, but you can add and subtract rates of working.

Let's express everything in terms of rate. First of all, a few variables
Rm = rate of one man working
Rw = rate of one woman working
Rc = rate of one child working

The first scenario tells us the combined rate of 2 men and 5 women is to do one piece of work in 8 days, which is a rate of (one piece of work)/(8 days) = 1/8, so
2Rm + 5 Rw = 1/8

The second scenario tells us the combined rate of 2 men and 7 children is do one piece of work in 12 days, which is a rate of (one piece of work)/(12 days) = 1/12, so
2Rm + 7Rc = 1/12

We want to relate women to children directly, so we need to eliminate men --- subtract the second equation from the first:

(2Rm + 5 Rw = 1/8)
-(2Rm + 7Rc = 1/12)
5Rw - 7Rc = 1/8 - 1/12 = 1/24

There's a problem here. We have two variables, and only one equation, and it's mathematically impossible to determine unique values for the two variables if there's only one equation. Are you sure the original problem doesn't give more information of some kind? Something about the problem as stated is not enough to work the problem through to completion. What is the source of this question?

I would be happy to help, but I think something is missing from the question.

Mike