Dividing a Whole Number by a Unit Fraction

4 ÷ 1/3 asks how many thirds fit into 4.

Count them

Each whole one holds 3 thirds. Four wholes hold 4 × 3 = 12.

The last rule to break

Division used to make numbers smaller. Here 4 ÷ 1/3 = 12, which is bigger.

Why it grows

Small pieces fit into something many times. The smaller the piece, the more fit.

How many small pieces fit

3 ÷ 1/4 asks how many quarters fit into 3 wholes. Each whole holds 4 quarters, so three wholes hold 12. The answer is 12 — larger than 3, and that is exactly right, because you are counting small pieces.

Why the answer grows

Dividing by a number less than one always gives an answer bigger than what you started with. The old rule that division makes things smaller assumed you were dividing by whole numbers greater than one. It has expired, exactly as the multiplication rule did.

Check by multiplying

12 × 1/4 = 3, which returns the number you started with. Multiplication checks division here just as it always has. This check is especially valuable when the answer's size feels wrong, because it settles the matter without argument.

A real situation

You have 3 metres of ribbon and need pieces a quarter of a metre long. How many pieces? 3 ÷ 1/4 = 12. Framing it as cutting a real length into real pieces makes the growing answer obviously correct rather than merely calculated.