This is a question that is somewhat inclined more towards statistics rather than mathematics, but because we can't consider it to be of a pure statistical nature, I believe this is the right place to ask this question.
In the city of New York, there are only four kinds of kinds of cars namely Alpha, Beta, Gamma, Omega.
The following table gives the individual percentage increment in the number of cars from $2017$ to $2018$. $$ \begin{array}{cc} \text{Alpha}& 8.88\text%\\ \text{Beta}&9.96\text%\\ \text{Gamma}& 12.44\text%\\ \text{Omega}& 11.68 \text%\\ \end{array} $$
It is also know that in the year $2017$ none of the four varieties of cars had a share less than one-sixth of the market share. Further we are also told that the overall percentage increase in the number of cars in New York from $2017$ to $2018$ is exactly $11\text%$. We are asked to find the minimum possible percentage share of Gamma and Omega vehicles in $2017$
I first calculated the maximum increment possible for the city and that came out to be around $11.29\text%$ Also we can see that we can't afford to have much of Alpha and Beta types to reach the desired $11 \text%$ Someone we need to tweak the $11.29\text%$ figure by making more of Omega. But I can't find any intuitive way to get through that. Also making equation would lead to a four variable single equation which will also prove difficult to tackle. Kindly help me getting through this question.
This is a classic constrained optimization problem. Imagine there is $1$ car in $2017$ as the car population will divide out. Let $a,b,g,w$ be the fraction of a car that each kind represents. You have constraints $a,b,g,w \ge \frac 16$. There are $1.11$ cars in $2018$. You can write a linear equation to reflect that the increases must add to $0.11$. You want to minimize $1.1244g+1.1168w$ subject to the constraints.