Aptitude: Time and Work Problem - How this formula works?

Question

If 5 men can color 50-meter long cloth in 5 days, in many days 4 men can color a 40-meter long cloth?

Quicker method:

Apply formula: M1 * D1 * W2 = M2 * D2 * W1

5 * 5 * 40 = 4 * D2 * 50

D2 = 1000/200 = 5 Days

This formula seems simple but I can't understand it intuitively. It would be helpful if you explain it more intuitively of how the formula works.

Answer 1

You need to think about worker efficiency. How do you define that? How many meters of cloth can one man color in one day. Let's use $E$ for this number, which you assume is the same for every worker. Then the amount of cloth $W$ that $N$ workers can color in $D$ days is $$W=E\cdot N\cdot D$$ You can write the efficiency as $$E=\frac W{N\cdot D}$$ Now use the same efficiency in both cases: $E=E$ implies $$\frac {W_1}{N_1\cdot D_1}=\frac {W_2}{N_2\cdot D_2}$$ or $$W_1\cdot N_2\cdot D_2=W_2\cdot N_1\cdot D_1$$

Answer 2

This is just unitary method.

Taking the Same variables as you took,

$M_1$ men can paint $W_1$ m wide cloth in $D_1$ days

So each can paint $\dfrac{W_1}{M_1\times D_1}$ m of cloth in one day. Let's call that $E$ (efficiency of each man per day)

So, efficiency per day of $M_2$ men is $E\times M_2$

And now you just have to divide width of cloth by total efficiency per day to get number of days required: $$D_2=\frac{W_2}{E\times M_2} \\= M_1\times D_1 \times W_2= M_2 \times D_2 \times W_1$$

Answer 3

The most intuitive way of understanding it is in the common fashion where we refer to "man-days".

A particular job can be done in a certain amount of time by a certain number of men.

To make the job finish quicker you need more men working together. (more men means fewer days to do a job)

If you employ fewer men you'll take longer to do the same job (fewer men means more days to do a job).

I hope you can now see how the composite unit "man-day" is helpful. It captures both of these factors. $1$ man-day is one man working for one day and is like $1$ unit of labour.

So let's employ this notion in an intuitive fashion first before touching the formula.

The first scenario has $5$ men working $5$ days, which is $25$ man-days. This amount of labour accomplishes colouring $50$ metres of cloth.

Which means colouring $1$ m of cloth would take $\frac{25} {50} =0.5 $man-day.

Colouring $40$ m of cloth would require $40 \times 0.5 = 20$ man-days.

Since you have a $4$ man team, they need to work for $\frac{20} {4} =5 $ days.

Now lets look at the formula. The product $M_1D_1$ means the number of man-days in one scenario while $M_2D_2$ is the number of man-days in a second scenario. Since these are units of labour, let's call them $L_1$ and $L_2$ respectively. So $L_1=M_1D_1$ and $L_2=M_2D_2$.

Now, the labour you put in will be directly proportional to the work you accomplish. So let's say your scenario $1$ accomplishes work $W_1$ (this can be something like colouring a certain length of cloth or building a certain number of houses, etc.) while scenario $2$ accomplishes work $W_2$.

By the law of direct proportion, you have that $L_1 = kW_1$ while $L_2 = kW_2$ where $k$ is a constant of proportionality. You can eliminate this constant by dividing the two equations: $\frac {L_1} {L_2} = \frac {W_1} {W_2}$.

Rearranging gives you $L_1W_2 = L_2W_1$.

Finally, substituting back the original expressions for labour in terms of man-days, we get: $M_1D_1W_2 = M_2D_2W_1$, which is exactly what you were asking about.

Note that I started with the intuitive explanation first, because that formula you're expected to apply is not something I would consider immediately intuitive. Sometimes, doing things from first principles is far preferable to blindly applying formulas. But at least now you know how it's derived (or at least I hope so).