A mason employed a certain number of workers to finish constructing a wall in a certain number of days.

Question

But soon he realized that the work would get delayed by ¼ of the time. He then increased the number of workers by a third and they managed to finish the work on schedule. What percentage of the work had been finished by the time the new labor joined?

I know this is the most hated part here at SE, but please provide some hints to begin with such problems. If the workers were 3K, and they took 4 days to complete the work, total units of work would have been 12K. Now, when mason realizes it will take 1 day more, what exactly is that point when he realizes this, so that I can compute what percentage of work was done till then.

Answer 1

We shall assume that the mason's initial error was due to a misjudgement and that the workers are always productive at a fixed rate.

Let $x,y,p$ be the initial number of workers, the desired completion duration in days, and the required percentage, respectively.

workers $(w)$	days $(d)$	jobs $(j)$
$x$	$\displaystyle\frac54y$	$1$
$x$	$\displaystyle\frac p{100}\left(\frac54y\right)$	$\displaystyle\frac p{100}$
$\displaystyle\frac43x$	$\displaystyle y-\frac p{100}\left(\frac54y\right)$	$\displaystyle\frac{100-p}{100}$

The joint proportionality among $w,d$ and $j,$ is such that $\displaystyle\frac{w_id_i}{j_i}$ has a fixed value.

Thus, $$\frac{x\left(\frac54y\right)}1=\frac{\left(\frac43x\right)\left(y-\frac p{100}\left(\frac54y\right)\right)}{\frac{100-p}{100}}\\p=20.$$

Answer 2

Let us assume $4t$ to be the expected time. Initial labor is $3m$.(without loss of generality). So $3m$ people actually take $5t$ time, so work $W=15mt$ (units).

Let $x$ be the required fraction . Then $15xmt$ of the work is done in $5xt$ time by $3m$ people .

He employs $4m$ total people now. So $4m$ people take $(4t-5xt)$time for $15mt(1-x)$ work as the work is now done in $4t$ time.

So, $$4m(4t-5xt)=15mt(1-x)$$ On solving yields $x=0.2$