A Fraction of a Fraction

1/2 of 1/3 asks for half of a third. The word "of" means multiply.

Cutting twice

Cut a whole into thirds, then cut each third in half. That makes sixths.

The rule this gives

1/2 × 1/3 = 1/6. Multiply the tops, and multiply the bottoms.

The answer is smaller than both

A piece of a piece is smaller than either. That is why the product shrinks.

Of means multiply

Half of 2/3 is written 1/2 × 2/3. The word "of" signals multiplication whenever it sits between a fraction and a quantity. Half of 10 is 5, and that is 1/2 × 10 — the same structure with a whole number.

Cutting twice

Shade 2/3 of a rectangle. Now cut the whole rectangle the other way, in half, and take half of the shaded part. You end up with 2 pieces out of 6: 1/3. The double cut is what produces the new denominator — 2 cuts times 3 cuts gives 6 pieces.

The rule and where it comes from

Multiply the numerators and multiply the denominators: 1/2 × 2/3 = 2/6 = 1/3. The denominators multiply because the two cuts combine to make that many pieces; the numerators multiply because you take that many of them.

The answer is smaller than both

A fraction of a fraction is smaller than either, because you are taking part of something that was already part of a whole. If your answer is bigger than what you started with, the operation went the wrong way. This check is quick and dependable.