In a field, two workers are planting trees. After sometime, a third worker is added and the number of trees planted becomes half as large. How many trees can the second worker plant as a percentage of the number of trees planted by first worker if it is given that efficiency of second worker is $\frac{1}{3}$ of $1st$ and $3rd$ worker combined.
My solution approach :-
Let the first worker be $W_{1}$ who can plant $x$ tress in $1$ unit of time.
Let the second worker be $W_{2}$ who can plant $y$ tress in $1$ unit of time.
Let the third worker be $W_{3}$ who can plant $z$ tress in $1$ unit of time.
Let $W_{1}$ and $W_{2}$ be working together for $t$ units of time before $W_{3}$ joined and hence the number of trees planted by $W_{1}$ and $W_{2}$ will be $(x+y)t$.
After $t$ units of time, $W_{3}$ joined them and all three together worked for next $T$ minutes and hence the number of trees planted by $W_{1}$ and $W_{2}$ and $W_{3}$ will be $(x+y+z)T$.
Total time taken = $t+T$
Total trees planted = $(x+y)t + (x+y+z)T$
As per question (the number of trees planted becomes half as large after $W_{3}$ joined $W_{1}$ and $W_{2}$);
$(x+y)t + (x+y+z)T = 1.5(x+y)t$
$\Rightarrow 2(x+y+z)T = (x+y)t$
Efficiency of $W_{1}$ will be $x$.
Efficiency of $W_{2}$ will be $y$.
Efficiency of $W_{3}$ will be $z$.
As per question (it is given that efficiency of second worker is $\frac{1}{3}$ of $1st$ and $3rd$ worker combined);
$3y = (x+z)$
After this I am not able to think how can we get to remove the variable $t$ and $T$ from the problem. Can someone please help me with this? I've been stuck with this for a while.
Thanks in advance !
Let trees planted by first worker be $x$, by second be $y$ and by third be $z$.
It is given that "a third worker is added and the number of trees planted becomes half as large" thus we can make this equation.
$$ \frac{x+y}{2}=z $$
and. Now see for the second equation the question is asking "How many trees can the second worker plant as a percentage of the number of trees planted by first worker if it is given that efficiency of second worker is 1/3 of 1st and 3rd worker combined" which means it is asking that if we time is same and they work together then percent of B's work in term of A.
So,
$$ \frac{x}{t} + \frac{z}{t} = \frac{3y}{t} $$
Cancelling all t from denominator,
$$ x+z=3y $$
Now first of all let us reduce both equation in terms of $x$ and $y$.
Substitute value of z from equation $1$ in equation $2$
You get $3x=5y$
$$ \frac{y}{x} = \frac{3}{5} $$
Now just multiply 100 by both sides to get the answer.