7.2 Change and Rates
Key Takeaways
- Absolute change is new minus old in the original unit; relative change divides that difference by the starting value
- A gap between two percents is percentage points; percent change divides that gap by the starting percent
- Average rate is total change in the dependent quantity divided by total change in the independent quantity
- Steepness of a graph is that rate: a steeper distance-time segment means a higher average speed
- Curvature means the rate itself is changing: the graph bends up when successive increases grow, and flattens when they shrink
Change has a size and a base
The QL subdomain Change and rates asks you to distinguish absolute from relative change, quantify differences, compute average rates, read steepness as rate, and read curvature as a changing rate. QL remains 50 multiple-choice questions in AQL, calculator-free, and not MAT calculus. Independent OpenExamPrep rate practice uses faculty enrolment, electricity tariffs, and residence-to-lecture travel. It does not reprint confidential booklet items.
Absolute change is new minus old, in the same unit as the quantity: students, kilowatt-hours, kilometres, rand. Relative change is that difference divided by the starting value, usually written as a percent. The starting value is the base. Change the base, and the same rand gap becomes a different percent.
| Language | Calculation | Unit |
|---|---|---|
| Absolute change | new − old | original unit |
| Relative change | (new − old) / old | fraction or percent |
| Percentage points | later % − earlier % | percentage points |
| Percent change of a percent | (later % − earlier %) / earlier % | percent |
| Average rate | (change in dependent) / (change in independent) | a per unit |
Enrolment: 120 people is not 120 percent
Worked example. A faculty first-year roll moves from 800 to 920 students.
Absolute change: 920 − 800 = 120 students. Relative change: 120 / 800 = 12 / 80 = 3 / 20 = 0.15 = 15%.
Saying “enrolment rose 120%” would mean the roll had more than doubled, to 800 + 800 = 1600, which is not 920. Saying “it rose 920%” copies the new headcount as a percent. Interpret back: the faculty added 120 students, which is a 15% rise on the 800.
If a later stem asked for the rise relative to 920, that would be a different base: 120 / 920 = 12 / 92 = 3 / 23 ≈ 13.0%. The absolute gap is still 120 students. Only the percent moved.
Percentage points are not percent change
Worked example: module pass rates. A campus report shows 64% of a first-year module passing in 2025 and 71% passing in 2026.
Arithmetic gap: 71 − 64 = 7 percentage points. Relative change of the pass rate: 7 / 64. 7 ÷ 64: 64 × 0.10 = 6.4; remainder 0.6; 0.6 / 64 ≈ 0.009; total ≈ 0.109 = 10.9%.
Calling the move “a 7% increase” is the trap. A 7% rise on 64% would be 0.07 × 64 = 4.48 percentage points, landing near 68.5%, not 71%. The board moved 7 percentage points, which is about an 10.9% relative rise on the 2025 pass rate. Chapter 6 trained the same distinction on occupancy; here the job is to keep using it when the story is a rate of change, not a ratio phrase.
Tariffs: two quantities can change at once
Worked example: residence electricity. Use rises from 120 kWh to 150 kWh. The teaching tariff in the item is R2.50 per kWh.
Absolute use change: 150 − 120 = 30 kWh. Relative use change: 30 / 120 = 25%. Absolute cost change at a fixed tariff: 30 × 2.50 = R75. Old cost 120 × 2.50 = R300; new cost 150 × 2.50 = R375; 375 − 300 = R75.
Now let the tariff itself rise from R2.50 to R2.80 while use stays 120 kWh. Absolute tariff change: R0.30 per kWh. Relative tariff change: 0.30 / 2.50 = 12%. New cost: 120 × 2.80 = R336. Absolute cost change: 336 − 300 = R36, which is 12% of R300, matching the tariff rise because use was held fixed.
If both use and tariff rise — 150 kWh at R2.80 — new cost = 150 × 2.80 = R420. Absolute cost change from R300 is R120. Relative cost change: 120 / 300 = 40%. That 40% is not 25% + 12%. Combining a 25% use rise with a 12% price rise is 1.25 × 1.12 = 1.40, a 40% cost rise. Adding the percents to 37% is the trap.
Average rates and travel time
An average rate of change is total change in the dependent quantity divided by total change in the independent quantity. Speed is distance per time. A tariff is rand per kilowatt-hour. Do not average the two segment speeds unless the story’s times (or distances) actually make that mean equal to total distance over total time.
Worked example: walk to campus. A student walks 2.4 km in 40 minutes. Time in hours: 40 / 60 = 2/3 h. Average speed: 2.4 ÷ (2/3) = 2.4 × 3/2 = 3.6 km/h. Check: 3.6 × (2/3) = 2.4 km. A trap is 2.4 / 40 = 0.06 km per minute, then labelling that 0.06 as km/h. Another trap is 40 / 2.4, which inverts the rate.
Worked example: shuttle. 9 km in 15 minutes. 15 minutes = 0.25 h = 1/4 h. 9 ÷ (1/4) = 36 km/h. 9 / 15 = 0.6 km per minute, and 0.6 × 60 = 36 km/h. Copying 9 or 15 as a speed in km/h skips the conversion.
Worked example: two legs, equal distance, unequal time. A minibus covers 18 km in 20 minutes, then another 18 km in 40 minutes. First-leg speed: 18 / (20/60) = 18 / (1/3) = 54 km/h. Second-leg speed: 18 / (40/60) = 18 / (2/3) = 18 × 3/2 = 27 km/h. Arithmetic mean of 54 and 27 is 40.5 km/h. That is not the trip average. Total distance: 36 km. Total time: 60 minutes = 1 h. Average speed: 36 km/h. The second leg lasted longer, so it must pull the average below 40.5. Interpret back: the minibus averaged 36 km/h for the whole 36 km, even though one stretch was 54 km/h.
Steepness is rate; curvature is a changing rate
On a graph of dependent versus independent, the steepness of a segment is the average rate on that interval. A steeper distance–time segment means more kilometres per hour. A horizontal segment means rate zero: enrolment unchanged, tank not filling, shuttle stopped.
Worked example: enrolment graph.
| Year | First-years |
|---|---|
| 2023 | 800 |
| 2024 | 840 |
| 2025 | 920 |
| 2026 | 920 |
2023 to 2024: +40 students in 1 year → 40 students per year. 2024 to 2025: +80 students per year — the segment is steeper. 2025 to 2026: +0 — the graph is flat; the rate dropped to zero.
Curvature is the story of those successive rates. From 2023–24 to 2024–25 the yearly increase grew (40 then 80), so the graph bends upward: the rate of change increased. From 2024–25 to 2025–26 the yearly increase collapsed (80 then 0), so the graph bends toward horizontal: the rate of change decreased. QL does not ask for a second-derivative formula. It asks whether the rises are getting bigger, getting smaller, or staying the same.
Worked example: tank graph. A residence tank plot of volume against time is shallow, then steep, then shallow again. Filling is slow, then fast, then slow. Average rate on each interval is still (litres later − litres earlier) / (hours later − hours earlier). The bend tells you the filling rate is not constant. A straight line through the same end-points would hide that changing rate.
A method
- Write new − old with a unit. That is absolute change.
- Divide by the stated base. That is relative change.
- If both numbers are already percents, name the gap in percentage points before you divide.
- For a rate, divide by the time, distance, or other independent span, converting minutes to hours when the option is per hour.
- On a graph, compare segment slopes, then say whether those slopes are rising or falling.
Traps
- Calling +120 students a 120% rise
- Calling a 7 percentage-point pass-rate move a 7% rise
- Adding 25% and 12% instead of compounding use and tariff
- Averaging two speeds when times differ
- Reading a flattening curve as a constant rate
- Using a calculator
Independent OpenExamPrep QL rate teaching is that checklist in higher-education contexts: the last number must still be a student count, a kilowatt-hour, or a kilometre per hour, not a free-floating percent.
Faculty enrolment rose from 800 to 920 students. The absolute change and the relative change on the 800 are:
A module pass rate moved from 64% to 71%. The change is best described as:
A campus shuttle covers 9 km in 15 minutes. Its average speed is: