Mini and Vinay are quiz masters preparing for a quiz. In x minutes,
Mini makes y questions more than Vinay. If it were possible to reduce
the time needed by each to make a question by two minutes, then in x
minutes mini would make 2y questions more than Vinay. How many
questions does Mini make in x minute?1] 1/4[ 2(x+y) – ( 2 x^2 + 4 y^2 )^1/2 ]
2] 1/4[ 2(x-y) – ( 2 x^2 + 4 y^2 )^1/2 ]
3] Either option 1 or 2
4] 1/4[ 2(x-y) – ( 2 x^2 – 4 y^2 )^1/2 ]
My attempt:
Let Mini and Vinay take M and V minutes to frame a question respectively.
Mini takes M mins to frame 1 question.
Therefore Mini takes $x$ mins to frame $\frac{x}{M}$ questions.
Vinay takes V mins to frame 1 question.
Therefore Vinay takes $x$ mins to frame $\frac{x}{V}$ questions.
ATQ:
$$\frac{x}{M}=\frac{x}{V}+y \tag 1$$
Multiplying the above equation by 2 yields:
$$\frac{2x}{M}=\frac{2x}{V}+2y \tag{1a}$$
Now, in the second scenario:
Mini takes (M-2) mins to frame 1 question.
Therefore Mini takes $x$ mins to frame $\frac{x}{(M-2)}$ questions.
Vinay takes V-2 mins to frame 1 question.
Therefore Vinay takes $x$ mins to frame $\frac{x}{(V-2)} $ questions.
Therefore ATQ:
$$\frac{x}{M-2}=\frac{x}{V-2}+2y \tag 2$$
Equating 2y from $(1)a$ and $2$ yields:
$$\frac{x}{M-2}-\frac{x}{V-2}=\frac{2x}{M}-\frac{2x}{V}$$
$$\frac{1}{M-2}-\frac{1}{V-2}=\frac{2}{M}-\frac{2}{V}$$
$$\frac{V-2-M+2}{(M-2)(V-2)}=\frac{2V-2M}{MV}$$
$$\frac{V-M}{MV-2M-2V+4}=\frac{2V-2M}{MV}$$
$$(V-M)MV=2(V-M)(MV-2M-2V+4)$$
$$MV=2MV-4M-4V+8$$
$$MV-4M-4V+8=0$$
What is the other equation?
If someone can tell us any other shorter method. That would be very helpful.But please do tell the other equation that would solve the problem.Thanks.
Edit: I have done as Rohan has mentioned in the comments.We basically have to find $\frac{x}{M}$.I have arrived at an answer which is close to the actual answer but not the actual answer.Please help me with this. (It would be very helpful if someone poits out my fault and works in the same direction as I had started).I would appreciate if someone does this in a concise way(You don't need to do it my way in this case).I haven't written this in mathjax as it would be very time consuming. Thanks.
Best Answer
If in $x$ minutes, Mini makes $a$ questions, then, Mini makes one question in $\frac{x}{a}$ minutes.
In $x$ minutes vinay makes $a-y$ questions, then, Vinay makes a question in $\frac{x}{a-y}$minutes.
Then, if Mini makes a question in $\frac{x}{a}-2$ minutes and Vinay in $\frac{x}{a-y}-2$ minutes, it is given that in $x$ minutes Mini makes $2y$ questions more.
Basically we want to find $a$ in the question. This can be solved by taking: $$\frac{x}{\frac{x}{a}-2}-\frac{x}{\frac{x}{a-y}-2}=2y$$ Solving the required quadratic in $a$, gives us one of the correct options as: $$\boxed{a= \frac{2(x+y)-\sqrt{2x^2+4y^2}}{4}}$$