Time and Work. How much work does C do per hour? 
A, B and C need a certain unique time to do a certain work. C needs 1
  hours less than A to complete the work. Working together, they require
  30 minutes to complete 50% of the job. The work also gets completed if
  A and B start working together and A leaves after 1 hour and B works
  for a further 3 hours. How much work does C do per hour? 
(A)16.66% 
(B)33.33% 
(C)50% 
(D)66.66%

My attempt:
Let the total work be 100 units.
Let the work done by A,B and C be a units/hour,b units/hour,c units/hour respectively.
Let the time taken by A alone to complete the work be t hours.
ATQ:
\begin{align*}
(a+b+c) \cdot \frac{1}{2} & =50 \tag{1}\\
(a+b) \cdot 1+b \cdot 3 & =100 \tag{2}\\
c \cdot (t-1)& =100 \tag{3}\\
at & =100 \tag{4}
\end{align*}
Please help me solve these equations. When I am solving it is getting cumbersome. 
Also if someone tells us some other way of solving, that would be helpful as well. Thanks.
 A: Let us assume, they take $a, b,$ and $c$ hours to finish the job individually, respectively. Note the following points:


*

*They complete half the job in half an hour $\implies$ full job is completed in a hour. Hence, $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$

*Also, A takes 1 hour more time than C. So, $a= c+1\implies \frac{1}{a}=\frac{1}{c+1}$

*If A works for a hour, and B works for four hours, then $\frac{1}{a}+\frac{4}{b}=1 \implies \frac{1}{b}=\frac{1-\frac{1}{a}}{4}=\frac{1-\frac{1}{c+1}}{4}=\frac{c}{4+4c}$.
Now, solving for $c$ gives us the quadratic: $3c^2-4c-4=0 \implies c=2, c= -\frac{2}{3}$. As, $c >0$, we have, the rate of work C does $=0.5$.

Solving by your method: $a =\frac{100}{t}, b = \frac{100-a}{4}=\frac{25t-25}{t}$. Then, we get, $$a +b + c =100 \implies 100+25t-25 + ct = 100t \implies t = \frac{75}{75-c}$$
Putting this in Eq.(3), we get, $$\frac{100}{c}=\frac{c}{75-c}\equiv 7500-100c = c^2 \equiv c^2+100c-7500=0 \implies c = \frac{1}{2}, -\frac{3}{2}$$
Rejecting the negative solution, we have $c =\frac12$, the same answer as obtained earlier. 
No mistake in your approach!
A: Let the total work be 1, or 100%.
Let the efficiency of A,B and C be a,b and c(percentage of work be done in an hour)

(1) $\frac{1}{c}$ + 1 = $\frac{1}{a}$

(2) $(a+b+c)\cdot\frac{1}{2}$ = $\frac{1}{2}$
(3) $(a+b)\cdot1 + b\cdot3 = 1$

from (3) you get

a + 4b = 1
b = $\frac{1-a}{4}$

put it into (2)

a + $\frac{1-a}{4}$ + c = 1
4a + 1 - a + 4c = 4
a = $\frac{3-4c}{3}$

put it into (1)
$\frac{1}{c} + 1 = \frac{3}{3-4c}$
$\frac{1+c}{c} = \frac{3}{3-4c}$
$4c^2 + 4c - 3 = 0$
$4c^2 + 4c + 1 = 4$
$(2c + 1)^2 = 4$
2c + 1 = 2 (c can not be negative)
c = 0.5
The final answer is C 50%.
A: Let $A$, $B$ and $C$ be the rates measured in an amount of work per hour $\left(\frac{work}{hour}\right)$ at which workers $A$, $B$ and $C$ accomplish work. If $C$ is the amount of work worker $C$ can do in one hour and $1$ ($100\%$) is the amount of work of the entire job, then $t$ is the time it takes him to bring it to completion: $t=\frac{1}{C}\cdot\frac{\text{work}}{\text{work/hour}}$. We also know that it takes worker $A$ one hour longer to complete the job than it takes worker $C$: $A(t+1)=1 \implies A\left(\frac{1}{C}+1\right)=1$. Another condition that we have is that three of them together require $30$ (or $\frac{1}{2}$ of an hour, don't forget that time is masured in hours here) minutes to complete $50\%$ of the job: $\frac{1}{2}(A+B+C)=\frac{1}{2} \implies (A+B+C)=1$. It is aslo stated that the work also gets completed if worker $A$ and worker $B$ start working together and worker $A$ leaves after $1$ hour and worker $B$ works for a further $3$ hours: $1\cdot(A+B)+3B=1 \implies A+4B=1$. We're all set to go now. Our conditions:
$$
A\left(\frac{1}{C}+1\right)=1\\
A+B+C=1\\
A+4B=1
$$

Solution:

$$
A\cdot\left(\frac{1}{C}+1\right)=1 \implies A=\frac{C}{1+C},\\
A+4B=1 \implies B=\frac{1-A}{4} \implies B=\frac{1}{4(1+C)},\\
A+B+C=1 \implies \frac{C}{1+C}+\frac{1}{4(1+C)}+C=1 \implies C^2+C-1=0 \implies \\
C_{1}=-\frac{3}{2}\text{ (discarded)}\\
C_{2}=\frac{1}{2}=\frac{1\text{ work}}{2\text{ hour}}≡50\%
$$
Answer: $C$ does $50\%$ of work per hour.
A: Ok I'm going to check just options so if you are looking for equations this is not your answer but question can be solved in this way too...
Let's take option B , i.e C does 33.33 % work in one hours
That means for 100% it requires 3 hours which means
A requires 4 hours for 100% work to be done.
That means per hour A does 25% work
Now ATQ for they all take 1 hour together to complete 100% work this means B will do 100-25-33.33 = 41.66% of work in 1 hour
Now verify this with another statement i.e A+4B=100 which shows that B is far more than required
Thus C should be more.
Now check for 50% work by C in 1 hour and everything fits perfectly.
PS: this method may seem cumbersome but it just require basic mathematics and no paper and pen is needed. Thus saves lot of time.
