Count the portions that fit
Dividing by a fraction can mean asking how many portions of that size fit into an amount. If you have 3/4 litre of juice and each cup holds 1/4 litre, you can fill three cups: 3/4 ÷ 1/4 = 3.
The answer counts cups. It does not mean you have made three litres of juice. Keeping track of what the answer counts makes the calculation much easier to understand.
Use matching pieces to see it
Picture three quarter-litre portions lined up. Each cup takes one portion, so three cups use the whole amount. OpenStax models fraction division in this way, by asking how many copies of one amount fit into another.
Now change the cup to 1/2 litre. Your 3/4 litre fills one full cup and half of another. That is 1½ half-litre portions: 3/4 ÷ 1/2 = 3/2.
Why the reciprocal rule helps
The reciprocal of a non-zero fraction swaps its numerator and denominator. To divide by that fraction, multiply by its reciprocal. So 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2.
The first fraction keeps its order. You invert the divisor, the fraction you are dividing by. For another example, 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6. Neither a fraction’s denominator nor the divisor can be zero.
Why the answer can be larger
There are four quarter-litre portions in one litre. Counting smaller portions can give a larger number of portions, so 1 ÷ 1/4 = 4. This is why the idea that division must always make a number smaller fails here.
Compare that with taking a fraction of an amount. Multiplication and division answer different questions. If 6/4 and 3/2 look like different amounts, revisit equivalent fractions.