A mason employed a certain number of workers to finish constructing a wall in a certain number of days. 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.
 A: 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, $$\frac{x\left(\frac54y\right)}1=\frac{\left(\frac43x\right)\left(y-\frac p{100}\left(\frac54y\right)\right)}{\frac{100-p}{100}}\\p=20.$$
A: 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 $x=0.2$
