Consider the experiment in this link.

The experiment includes using a ruler as a lever, with an inflated balloon on one side and a balloon which is not inflated on the other.

The aim of the experiment is to show that air has mass.

I have seen many kids performing similar experiments.

But, if the air pressure inside the balloon is equal to that outside, then the buoyant force will cancel out the weight of the air inside the balloon, won't it?

I can think of at least four things going on in this experiment that need pointing out:


When you inflate a balloon by mouth, the air is warm: this makes the air inside the inflated balloon slightly lighter than the air it displaced
The air inside the balloon has 100% relative humidity at 37C, and condensation will quickly form on the inside of the balloon as the air inside cools down.
The air inside the balloon contains carbon dioxide, which has higher density than room air (molecular mass of 12+16+16 = 44 amu, vs oxygen at 32 amu and nitrogen at 28 amu - ignoring small isotopic effects, and ignoring Argon).
The pressure inside the balloon is larger than outside - this increases the density


So how large are each of these effects?


Warm air: 37C vs 20C results in drop in density of 0.945x (293 / 310) or -5.5%
Moisture: partial pressure of water at 37C is 47.1 mm Hg source which is about 0.061 atmospheres. Assuming that pressure is constant, this water (mass 18 amu) displaces air (mean mass 29 amu), so the density of the air decreases by 0.061 * (29 - 18) / 29 = 2.3%. If we allow the air outside the balloon to have 60% relative humidity (with saturated vapor pressure of 10.5 mm Hg), it would be slightly less dense than dry air (10.5*0.6/760*(29-18)/29 = 0.3%) making the net difference -2.0%. Note that much of this moisture will condense when the balloon cools down - little droplets will form on the inside of the balloon. With the air inside still saturated, its density will be 0.1% lower than on the outside; the net result amounts to 2.9% of the mass of the air in the balloon.
Carbon dioxide: the exhaled air has 4 - 5 % carbon dioxide source: wikipedia, with an equivalent drop in oxygen. The density of exhaled air is therefore higher than that of inhaled air by 0.045 * (44 - 32) / 29 = +1.9%
Pressure in the balloon: from this youtube video - time point 3:43 I estimate the pressure increase in the balloon at 23 mm Hg, resulting in an increase in density of 2.9%


Summarizing in a table:

factor      effect   at room T
temperature -5.5%       0.0%
moisture    -2.0%       2.9%
CO2          1.9%       1.9%
pressure     2.9%       2.9%
net         -2.7%       7.7%


A freshly inflated balloon will thus have only a slightly lower density than the air it displaced, because the temperature + moisture effect is greater than the other two.  After you wait a little while, the temperature will equalize and the density of the air inside the balloon will be greater - by 7.7%, with more than half of that not caused by the pressure in the balloon...

In summary: the experiment described in your link measures the difference in density between air in a balloon, and ambient air. Since the density of the air inside the balloon is higher than the density outside the balloon, one may conclude that the air inside the balloon has finite density. One may NOT conclude that the medium outside the balloon (which we believe to be "dry air") has any density at all - since nothing in this measurement tells us about the air outside the balloon.

If you did the experiment carefully with a balloon initially filled with warm air, and you allowed the air to cool down, you might be able to tell that the balance shifts - in other words, that there must be a change in the buoyancy experienced by the balloon as it cools down. THAT would be an experiment to demonstrate "air has mass" (volume of balloon decreases, and it experiences less buoyancy). From the experiment as described (popping the balloon), we learn that "exhaled air has mass". That is not the same thing.

If you used an air pump (balloon pump) to inflate the balloon, the first three components would go away and you are left with the difference due to the pressure only - 2.9% of the mass of the air in the balloon.
