Parts Smaller Than Wholes

Proper fractions — where the numerator is smaller than the denominator — represent everything less than one whole. They show up in measuring cups, time, and probability. Half a cup of sugar is 1/2. Fifteen minutes of an hour is 1/4. A 1-in-6 chance is 1/6.

Proper fractions feel more natural than improper ones because they map to real "part of a thing" situations. They also explain a small mystery of decimals: a proper fraction like 1/4 turns into a tidy 0.25, while 1/3 becomes 0.333..., repeating forever. Whether the decimal stops or repeats depends entirely on the fraction you started with.