Decimal division: use one group as the unit

Share 1.2 L of juice into 0.2 L cups. Imagine the cups: totals of 0.2, 0.4 and 0.6 L accumulate, reaching 1.2 L after six cups.

If the decimal point distracts you, count both amounts in tenths of a litre. You have twelve units and each cup holds two, so 12÷2=6. You have changed the counting unit, not the juice or cups.

This explains rewriting 1.2÷0.2 as 12÷2. Changing only the divisor to 2 would enlarge each cup tenfold and create a different problem.

Does division always make things smaller? The quotient here is 6, but not 6 L. It counts 0.2 L cups. Distinguish the original quantity from the number of groups.

For 1 L shared into 0.3 L groups, the quotient is 10/3: three full cups and 0.1 L left. The remainder is a third of one 0.3 L cup. The decimal 3.33 is an approximation, not the exact quotient.

A zero-sized cup cannot hold the juice, so division by zero does not define a group count. If the amount is zero and the cup size is positive, the count is zero.