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I remember back in school (some time ago) we were taught to solve problems such as the following:

If $6$ men can do $1/3$ of work in $10$ days then how many days would it take $4$ men to do $2/3$ work?

Our teacher constructed the following diagram to solve the problem enter image description here

and solved it as follows:

$$6\times 10\times \frac 23 = 4\times x\times \frac 13 \implies x=\frac 13$$

Now I am already familiar with the scenario on when to use cross-multiplication such as (when both the columns in the next row increase or decrease simultaneously) enter image description here

and I am also familiar when to use horizontal multiplication (When one column in next row increases and the other decreases or vice versa such as no of men and days example)

Coming back to the main question is there a key to remembering which column gets placed in the center or any other points to consider when constructing such a diagram ?

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You have a typo in your equation (right side should be 1/3 instead of 2/3), and also you didn't solve for $x$ correctly. – Ted Jun 1 '12 at 6:34
Thanks for pointing that out. – Rajeshwar Jun 1 '12 at 6:53
up vote 3 down vote accepted

I don't think there's much point in learning such ad-hoc tricks to solve particular forms of problems. Rather, one should understand how to translate the problem into algebra.
The basic idea is that the amount of work done is assumed to be proportional to both the number of workers and the time taken: thus $w = c m t$ for some constant $c$, where $w$ is the amount of work, $m$ the number of workers and $t$ the time. This is equivalent to $c = m t/w$. So if we have one data point with $m = 6$, $t =10$ and $w = 1/3$, and our new scenario involves $m=4$, $t=x$ and $w=2/3$, that says $\dfrac{6 \times 10}{1/3} = \dfrac{4 \times x}{2/3}$, which you then solve for $x$.

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Thanks this is an awsome method. Determin the propotionality and then construct an equation. – Rajeshwar Jun 4 '12 at 15:06

Instead of trying to remember what numbers to put in what columns in a diagram, it's better to just reason it out. 6 men can do 1/3 of the work, so it takes 18 men to do all of the work. Since the work happens in 10 days, the total work is 180 man-days of work. So, 2/3 of the work would be 120 man-days, which means it would take 4 men 30 days to do it.

The diagram you're talking about is just a compressed form of this reasoning.

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Thanks Ill keep this in mind – Rajeshwar Jun 1 '12 at 6:51

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