Make a cheapest battry airpalne
Not tough if u could find these things1. old picooz rc helicopter's broken tail part or a small pager motor or vibrator which u may get for free of cost in some mobile repair shops2. 3volts lithium "button battery''3. small propeller or just follow this " https://www.instructables.com/id/mini-airplane/ " to make a cheap best propeller from old cannow simple just follow this link to make a wonderful electric plane " http://scienceglitz.blogspot.com/2011/07/awesome-mod-for-walk-along-glider.html "
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At such a low price you will have to "find" a lot of things - Think about rubber band power at first.I assume your not expecting RC for that price.
Polystyrene foam is probably the best thing to use, and skin the wings with cling film to save weight.For power, the cheapest option is to re-wind a small 3V motor with some magnet wire, and connect it to a couple of rechargeable batteries (need to be able to supply a high current). This setup will get hot, is likely to wear out the motor, and is far less efficient than a properly designed motor, but will work for a while.
I think you're going to have trouble finding a motor and battery which can produce enough thrust to make up for their weight, especially at that price.Sorry...
At that price, I think you might be hard pressed to find a cheap enough motor, let alone some batteries and an airframe. What type of performance are you looking for?