Question: I'm currently a high school student trying to build an ultralight aircraft from scratch (I'm self taught and still learning from educational videos), but I don't have much savings (about RM 2.5k, which converts to roughly 587 USD), so I could only afford a small engine. 

Is it possible to achieve a certain propeller's efficient rpm by using gears while still using a small engine? Like those gearboxes found in cars to increase rpm.

Answer: 
The short answer is yes you can use some cheaply available engines to achieve flight however you are presumably asking about things like small car engines or motorcycle engines etc. The issue you have here is that these engines are [inherently different](https://aviation.stackexchange.com/questions/11527/what-are-some-of-the-differences-between-piston-engines-used-in-aircraft-and-aut/11528#11528) from a use case standpoint than your typical aircraft engine. Generally speaking, most car/motorcycle engines rev too high to be used in an aircraft, you actually want to lower the RPM, not raise it. You can read up on it [here](http://www.flyingmag.com/aircraft/do-car-engines-make-good-airplane-engines) and [here](https://www.reddit.com/r/askscience/comments/2fedpn/why_dont_propeller_driven_aircraft_use_automotive/) and [here](https://aviation.stackexchange.com/questions/11527/what-are-some-of-the-differences-between-piston-engines-used-in-aircraft-and-aut). You can pull it off with a stepper gear box ([like a turbo prop uses](https://aviation.stackexchange.com/questions/13837/given-the-same-engine-why-install-a-gearbox-on-a-turboprop-but-not-on-a-turbofa)) and it was even been done on [a piston plane back in the 80's by Mooney](https://en.wikipedia.org/wiki/Porsche_PFM_3200). The other problem you will have with car engines is cooling them, generally cars are watercooled, something that adds a lot of weight and is not ideal for flight. A two stroke or other similar air cooled motorcycle/scooter engine may be a better thing to use. 

Many small engines will produce more than enough horsepower to achieve flight. Keep in mind the [early piper cubs only had 40HP](https://en.wikipedia.org/wiki/Piper_J-3_Cub) which can be easily found in many 2 stroke engines. 

Your best bet may be to look for a [used VW bug engine](http://www.greatplainsas.com/) as they are very similar to aircraft engines and even [were used in some air frames](http://www.enginehistory.org/HOAE/Limbach.html) at one point in history. Due to the cars high production numbers the engines are fairly easy to come by.

You did not list a jurisdiction in your question but the regulations and legislation on this change from place to place, so you may need to see what exactly you are allowed to do in your jurisdiction in relation to home built and experimental aircraft.
Keep in mind that I'm very interested in airplanes, but I don't fly anything full scale, so others may have better advice.

There's a very simple, very light homebuilt called the Sky Pup. I'm not sure, but I think the plans are still being sold. Many of these have been built, and they have a good reputation. I've read that the guys who designed it were aero engineers. The materials are mostly wood and extruded polystyrene foam. The instructions tell you that anything more than 20 hp is too much. You can find out more at machnone.com (My only connection with the Sky Pup is that I bought a set of plans and have read about it here and there.) If you weigh less than 170 lbs, you might find this a good choice.
Reduction drives are common, especially with two stroke engines. Another approach is to use half of a VW engine, although some of the power will be wasted. These engines are much too heavy for a Sky Pup, though. There's a design called the Legal Eagle, though, which uses a 1/2 VW engine. 

New paramotor engines are probably far beyond your budget, but they might give you an idea of what can be done. They all have belt drives to reduce the propeller rpm. However, keep in mind that reduction drives can have big problems, unless they're engineered just right or have been tested and modified many times.

Unless you are very good at scrounging, or prices are MUCH lower where you are than they are here in the USA, your budget won't go far enough. Maybe there are some glider designs, though, that could be done with less than $1000 (US). 

Depending on how an "ultralight" is defined in your country, there may be a bunch of other designs that you can use. For instance, the Pietenpol Sky Scout, which is smaller than the famous Air Camper, is meant for 20 or 30 hp (not sure which). It wasn't known for climbing quickly, but I expect a modern, lighter engine would help with that. I mention it because it's an old, low tech design which may be cheaper to build.

There are MANY other choices out there. Whatever you choose, get some flight training before flying solo. And be really careful when you're building it.