Is it True that vehicles give more mileage when driven in 3rd gear?

Generally, no. If that were the case, they would have stopped developing automatic transmissions after the three-speed. Now there are four, five,six speed automatics. BMW and Mercedes have some seven speeds and the new Lexus LS460 has an eight-speed automatic.

The more gears, the better the performance in the car in many ways. Acceleration is usually faster because the engine is constantly in its power band. In most cars with the very popular 4 speed automatic, when you accelerate through all the gears, there is a big drop in RPM when shifting. On some cars, especially small engines or turbo engines, that drop is enough to take the car out of it's optimum power band, which is where the engine generates most of its power. It can have a big effect on acceleration times, unless its a car like a Corvette, where power is not an issue at any time.

More gears also help fuel economy at cruise speed by lowering engine RPM. When cruising at a steady 70mph, we don't need the power band of the engine, so it cruises along at a constant RPM. The less RPM being generated, the less fuel being burned. Example: My Mitsubishi Lancer has a 2.4 four cylinder that makes 160 hp. and has a four speed automatic. My wife has a Honda Accord with a 166hp four cylinder and a five speed automatic. At 70 mph cruise (with cruise control on) the Accord is turning 2500 rpm, while my Mitsubishi is turning 3000 rpm. The Accord is rated at 34 mpg highway, the Lancer only 28 mpg. And the Lancer is smaller. Other factors figure in (design, friction,etc.) but mainly it's the RPM drop on the Honda.

The only time driving in third would help is when climbing a long,steep grade. Even then, most cars now can sense when climbing a hill and automatically hold the car in third (or whatever gear is optimal)

Sorry for the long answer, but it's not a simple question.

Not necessarily... I presume you are referring to a manual transmission and probably a 5 spd. Depending on the speed you are travelling, the gear you are in is not what determines the fuel consumption.... it is your rate of acceleration and your engine speed (rpm). If you are looking to get the best fuel economy, change your gears at around 3,000rpm and don't have a cruising speed where the rpm are higher. Some cars have an arrow or light that comes on telling you when to change gears to get the best gas mileage.

yes. my husband drives only new cars. after it is broken in, the mileage steadily increases & levels off. he is a meticulous driver (no jackrabbit starts, slowly accelerates, slows down before stopping, drives a cold car slowly until needle move etc. follows the manual's instructions etc) he claims the car will act like the owner drives it. He is 70 and has had many new cars and gets 200-300,000 mi. he drives at least 100 mi per day, always standard trans.