One US tablespoon is exactly 3 teaspoons. The relationship holds in the American system, in the metric system where 15 ml is three times 5 ml, and in the imperial system — but not in Australia, where the tablespoon is four teaspoons.
About the units
Both units are defined through the US gallon: the tablespoon is half a fluid ounce and the teaspoon a sixth of one. In millilitres they are 14.7868 and 4.9289 respectively, and the ratio between them is exactly three because both derive from the same chain. The metric kitchen keeps the same structure with rounder numbers: matsked 15 ml, tesked 5 ml. The Australian exception is genuinely an exception rather than a variant — the Australian tablespoon was set at 20 ml when the country metricated, giving four teaspoons to the tablespoon and breaking a relationship that had held for centuries.
The exact factor
The factor is exactly 3, which makes this one of the few kitchen conversions that can be done confidently in the head. Two tablespoons is 6 teaspoons, and a third of a tablespoon is one teaspoon. Because 3 does not divide 16 evenly, the relationship upward to the cup is less clean: a cup is 16 tablespoons or 48 teaspoons, so a third of a cup is 5⅓ tablespoons — the awkward fraction that American measuring sets handle by including a dedicated ⅓-cup measure.
Where you meet this conversion
This is the conversion cooks do without a tool, usually when a measuring spoon is dirty or missing. It matters most when scaling a recipe down: a quarter of two tablespoons is 1.5 teaspoons, which is measurable, where a quarter of a tablespoon is not. Baking recipes frequently mix the two units in the same ingredient list — a tablespoon of oil next to a teaspoon of baking powder — and converting to a single unit before scaling avoids a class of arithmetic errors.