On: Can 1/3 and 1/3 = 2/6? It seemed so

[DigitallyFidget]

The issue is that it's ratios, not fractions. 1:3. You take 1:3 and 1:3 and it's still 1:3 or 2:6. You haven't changed the ratio for this at all by showing it as fractions, it's simply being represented and presented without correct context.

[TeMPOraL → DigitallyFidget]

Isn't this why you end up seeing these "silly" units like kg/kg in chemistry? So that, while the value is technically dimensionless, it doesn't get added to another dimensionless value (e.g. l/l) that's a ratio of values of a different dimension?

[romwell → TeMPOraL]

This hits the nail on the head:

1 (person at table A) / 3 (people at table A) can't be added to 1 (person at table B) / 3 (people at table B) without conversion of units.

[dwild → romwell]

And then it's easy to make them see how her answer was right in its own way by adding another unit, (person at table a+b and people at table a+b) and show why it may be harder to works like that for now.

I heard so many people complains that each year in maths they would essentially learn that everything they learned the year before was wrong... can we fix that please?!

[matt-attack → romwell]

Why? Each is a pure number without units. Since the units cancel in each term.

[mjevans → DigitallyFidget]

Another way of labeling this:

  : = Out Of
  A = Table A
  B = Table B
  E = Every Table Added (A + B)
  
  1:3 * A + 1:3 * B = (1:3 + 1:3) * E
  2 out of 6 among Every Table Added

[Wowfunhappy → DigitallyFidget]

Somewhat aside, but in grade school I recall learning that ratios were written [group-A]:[group-B], not [group-A]:[total]. So the girl-boy ratio at the table would be 1:2. Different for you?

[saagarjha → Wowfunhappy]

It really depends on the two things being compared. The ratio of boys to girls is 1:2, but the ratio of boys (implicitly: to the total) is 1:3.