Time And Work

Let ''W'' be the total work to be done ,

Let w1,w2 be the rates at which work is being done independently by KARAN and VIJAY respectively, t---------------Time taken;

FROM THE GIVEN DATA, (w1+w2)10 =W ; and

2.5w1+8.5w2 = W/2

=>8.5(w1+w2) - 6w1=W/2 ;

SUBSTITUTING w1+w2 VALUE , WE GET , w1=(3.5/60)W

                                                                                                                   =>TIME REQUIRED t = W/w1

                                                                                                                     =60/3.5

                                                                                                                     =17.142

Let the number of man-hours to complete the work be $W$ , and the numbers of hours for Karan and Vijan to complete the working alone be $k$ and $v$ respectively. Then, the portion of $W$ completed by Karan and Vijan in $1$ hour are $\frac{W}{k}$ and $\frac{W}{v}$ respectively. Therefore, we have:

$\begin{cases} \dfrac{10W}{k} + \dfrac{10W}{v} = W & \Rightarrow \dfrac{10}{k} + \dfrac{10}{v} = 1 &...(1) \\ \dfrac{2.5W}{k} + \dfrac{8.5W}{v} = 0.5W & \Rightarrow \dfrac{2.5}{k} + \dfrac{8.5}{v} = 0.5 &...(2) \end{cases} \\ \Rightarrow \begin{cases} (1)\times 8.5: & \dfrac{85}{k} + \dfrac{85}{v} = 8.5 & ...(1a) \\ (2) \times 10: &\dfrac{25}{k} + \dfrac{85}{v} = 5 & ...(2a) \end{cases}$

$\begin{aligned} (1a)-(2a): \quad \frac{60}{k} & = 3.5 \\ \Rightarrow k & = \frac{60}{3.5} = \boxed{17.143} \text{hours} \end{aligned}$

(Be warned, this approach is rather novel, and serves best as an explanatory tool rather than a practical approach to the problem).

Let W be the total work that needs doing, and let the units be 'workblocks.' So in other words, there are W 'workblocks' that need to be done. Let us then define K, where K the amount of work that Karan can accomplish per hour:

Karan's rate $=K\dfrac { workblocks }{ hour }$

Let us similarly define V as the amount of work that Vijay can accomplish per hour.

Vijay's rate $=V\dfrac { workblocks }{ hour }$

Now, we may begin to clothe the problem with some equations. First, we are told that working simultaneously, Karan and Vijay can complete the work to be done W in 10 hours. It is important to realise that both of them each work 10 hours. This understanding can be written thus:

$W$ $workblocks = 10$ $hours \cdot$ $K$ $\dfrac{ workblocks }{ hour } + 10$ $hours \cdot$ V $\dfrac{ workblocks }{ hour }$

While this is quite cumbersome, it's immediately obvious that the hour units cancel on the right hand side, leaving the unit 'workblocks'; this being the same unit on the left hand side, it is a little confirmation that we're on the right track, and that the equation makes sense. After we cancel the hours units and factorise, we are left with the much nicer equation, realising that the numerical coefficient of K and V is the amount of hours:

$W$ $workblocks = 10(K + V)$ $workblocks$

Secondly, we are told that if Karan works for 2.5 hours, and Vijay works for 8.5 hours, there is still half the work to be done. Phrased more usefully, Karan and Vijay have accomplished half the work to be done. We can describe this information thus:

$0.5W$ $workblocks = 2.5$ $hours \cdot$ $K$ $\dfrac{ workblocks }{ hour } + 8.5$ $hours \cdot$ V $\dfrac{ workblocks }{ hour }$

$W$ $workblocks = (5K + 17V)$ $workblocks$

Now, up until this point, we've done nothing but describe the problem mathematically. But that is essentially enough to solve the problem quite easily - we can simply write the two equations we've derived as two simultaneous equations, discarding the units as they're no longer necessary to aid our intuition:

$W = 10K + 10V$

$W = 5K + 17V$

We must ask ourselves now, what is it that we're trying to find? The question asks us for how many hours Karan would need, working alone, to complete all of the work to be done ( $W$ ). So, we want to know how 'many' workblocks per hour are necessary to equate $W$ workblocks. In other words, we want to find how many $K$ s are necessary to equate $W$ . In order to discover this, we eliminate the V to find $W$ in terms of $K$ :

$W = \dfrac{120}{7} K$

So in other words, it takes $\frac{120}{7}$ of $K$ to equate $W$ . It is thus immediately obvious that Karan takes $\frac{120}{7}$ hours to finish the work alone.

$\dfrac{120}{7}$ hours $\approx 17.142$ hours.

let x and y are the times to finish the work karan and vijay resp. so

he has given

1/x +1/y=1/10 from this i.e 10y+10x=xy

and 2.5/x + 8.5/y=1/2 from this 5y+17x=xy

from the above two eqns 5y=7x so x=(5/7)y substitue x value in

eqn1 you will get y=24 so x= (5/7)24 = 120/7= 17.142

Consider X is the total work and K & V be working time for Karan and Vijay respectively to complete X work independently.
Now working speed of Karan will be X/K and that of Vijay will be X/V.
As per given we have
(X/V)x10+(X/K)x10=X ---------(1) &
(X/V)x2.5+(X/K)x8.5=X/2 -----(2)
On Solving (1) and (2) we have
V=24 and K=240/14
i.e K=Time when Karan work alone=17.142 hours