Short Answer and Calculations $(80$ points total$)$

Show all your work to earn partial credit.

$(9$ points$)$ Luke Skywalker realizes that, due to his popularity, green lightsabers have become a normal good, with an income elasticity of demand of $0\mathrm{.}4.$ If Luke's income is $$50,000,$ and he buys $350$ green lightsabers, when each green lightsaber costs $$100$ a unit, answer the following questions:

a$.(4$ points$)$ If Luke's income increases by $50-$percent, from $$50,000$ to $$75,000,$ how many green lightsabers will he now purchase?

DO NOT USE THE NUMBERS FOUND IN PART A$.$ RE$-$WORK THE PROBLEM USING THE ORIGINAL VALUES $($each lightsaber is $$100$ and he buys $350).$ Coupled with this massive surge in popularity has come a price increase, where the own$-$price elasticity of demand for the green lightsabers are $-0\mathrm{.}5.$

b$.(5$ points$)$ If the price of green lightsabers increases by $50-$percent, to $$150$ a unit, how many green lightsabers will he now purchase?