What does a late penalty really cost?

A syllabus says late work loses 10% per day. It sounds precise, and it hides a question the sentence does not answer: 10% of what? The two common readings give different grades from the same work, and the gap between them grows with every day.

Two readings of the same sentence

Points off the maximum. Each day late removes a fixed slice of full marks, the same for every student. At 10% per day, that is ten percentage points a day:

late score = score − rate × days

Percent off your score. Each day removes a share of what the work earned, so the penalty scales with the work:

late score = score × (1 − rate × days ÷ 100)

Take work that would have earned 85%, handed in two days late at 10% per day. Under the first reading it keeps 65%. Under the second it keeps 68%. Same work, same sentence, three points apart.

Day by day

Here is the same 85% piece of work at 10% per day, from on time to ten days late:

The gap starts at 1.5 points on day one and adds 1.5 more each day, reaching 12 points by day eight. Then the first reading runs out: on day nine it would go below zero, so it stops at 0%, while the second reading still leaves 8.5%. By day ten both are gone.

Which reading is gentler

For any score below 100%, percent off your score always leaves more. The difference between the two readings is exactly

rate × days × (1 − score ÷ 100)

which is never negative, and is zero only for perfect work. At 85% that is 10 × 2 × 0.15 = 3 points after two days, which is the 65% against 68% above.

The readings also disagree about when work runs out. Under points off the maximum, a weaker piece of work reaches zero sooner: at 10% per day, 85% lasts until day nine but a 50% piece is worth nothing on day five. Under percent off your score every piece of work reaches zero on the same day, 100 divided by the rate, which is day ten at 10% per day, whatever it would have earned.

What it does to your course grade

An assignment only moves the course grade in proportion to its weight. Suppose the 85% work is worth 20% of the course. On time it would have contributed 17.0 points of course grade. Two days late it contributes 13.0 under the first reading or 13.6 under the second, so the lateness costs 4.0 or 3.4 points. Not handing it in contributes 0.

That last comparison is the one the arithmetic settles. As long as the late score is above zero, late work is worth more to the course grade than no work, and the calculator prints how much. Whether it is worth the time is a different question, and not one a calculator can answer.

Cutoffs, partial days and grace periods

Many syllabi stop accepting late work after a set number of days, and past that point a submission counts as a zero however good it was. The calculator's optional cutoff box models exactly that: enter the last day on which late work is accepted, and any later day scores zero.

Other details vary too much to assume. Some courses count any part of a day as a full day, some give a grace period of a few hours, some skip weekends, and some cap the total penalty. The calculator takes whole days and applies the rate you enter to each of them, and nothing else. Your syllabus wins on every one of these points; count the days its way before you type them in.

Every number above is checked by an independent test suite before it ships: the two formulas, the day-by-day table, the zero days, the gentler-reading identity and the course-grade figures are all reproduced in python and compared against the shipped JavaScript.