7.2 Change & Rates
Key Takeaways
- Absolute change is new − old in the original units; relative change is absolute change ÷ original value, usually written as a percent
- An average rate divides total change in one quantity by total change in another (for example km per hour, litres per day, rand per tutorial)
- On a graph, steeper segments mean larger rates for the same axes; compare rise over run, not only endpoint heights
- Curvature (a bend) signals a changing rate — speeding up, slowing down, or a non-constant increase — not a single fixed slope
Why change and rates matter on QL
The CEA QL construct on change and rates asks whether you can quantify how one quantity moves with another: fees over years, water over days, marks over study hours, or distance over time. On AQL you must separate absolute change from relative (percent) change, compute average rates, and read steepness and curvature on graphs — all without a calculator.
Earlier ratio work already used percent change. Here the emphasis shifts to rates as slopes and to recognising when a rate itself is changing.
Absolute change versus relative change
| Idea | Formula | Units / form |
|---|---|---|
| Absolute change | new − original | Same units as the quantity (rand, students, litres) |
| Relative change | (new − original) ÷ original | Pure number; × 100 for a percent |
| Relative decrease | (original − new) ÷ original when new < original | Percent of the starting value |
Worked example — residence fees: Fees rise from R8 400 to R9 660.
- Absolute change = 9 660 − 8 400 = R1 260
- Relative change = 1 260 ÷ 8 400
Simplify: divide numerator and denominator by 420 → 3 ÷ 20 = 0.15 = 15%.
Trap: dividing 1 260 by 9 660 (the new fee) gives a different percent and answers a different question. Unless the stem names another base, percent change uses the original.
Worked example — enrolment drop: A tutorial group falls from 160 to 120 students.
- Absolute change = 120 − 160 = −40 (a drop of 40)
- Relative decrease = 40 ÷ 160 = 0.25 = 25%
Saying “enrolment fell by 40%” would be wrong — it fell by 40 students, which is 25% of the start.
Average rates
An average rate is (change in quantity A) ÷ (change in quantity B) over an interval.
Distance / time: A shared taxi covers 180 km in 3 hours.
- Average speed = 180 ÷ 3 = 60 km/h
That average does not claim the speed was constantly 60 km/h — only that total distance over total time yields 60.
Volume / time: A campus tank holds 1 200 L on Monday morning and 400 L on Friday morning (4 days later), with no refill.
- Absolute drop = 1 200 − 400 = 800 L
- Average rate of use = 800 ÷ 4 = 200 L/day
Money / count: Printing costs rise from a R0 start to R96 after 24 pages at a flat per-page rate (ignore activation for this stem).
- Average rate = 96 ÷ 24 = R4 per page
Steepness of a graph = rate
When a graph plots dependent quantity vertically against independent quantity horizontally, steepness (rise ÷ run) is the rate for that segment.
Worked example — study hours vs quiz score (linear segments):
| Study hours | Quiz score (/50) |
|---|---|
| 0 | 20 |
| 2 | 28 |
| 5 | 34 |
- From 0 to 2 hours: rise = 28 − 20 = 8; run = 2; rate = 8 ÷ 2 = 4 score points per hour
- From 2 to 5 hours: rise = 34 − 28 = 6; run = 3; rate = 6 ÷ 3 = 2 points per hour
The first segment is steeper. A student who only compares endpoint heights (34 vs 28) misses that the rate slowed after hour 2. On QL, options often ask which interval shows the greater rate of improvement — answer with rise/run, not with “who ended higher.”
Load-shedding hours (bar or line reading): Week 1 = 14 h, Week 2 = 14 h, Week 3 = 7 h.
- Week 1 → 2: absolute change 0; rate of change 0 h/week (flat)
- Week 2 → 3: absolute change −7 h; relative to Week 2: 7 ÷ 14 = 50% decrease
Flat segments mean rate zero for that measured quantity, even if the absolute level is still high.
Curvature means a changing rate
A perfectly straight distance–time graph means constant speed. A curve that gets steeper means the object is speeding up (rate increasing). A curve that flattens means slowing down.
Worked example — cumulative water collected from a rain tank (litres vs hours):
| Hour | Cumulative litres |
|---|---|
| 0 | 0 |
| 1 | 10 |
| 2 | 30 |
| 3 | 60 |
Average rates hour-by-hour:
- Hour 0→1: (10 − 0) ÷ 1 = 10 L/h
- Hour 1→2: (30 − 10) ÷ 1 = 20 L/h
- Hour 2→3: (60 − 30) ÷ 1 = 30 L/h
The rate is rising each hour — the cumulative graph bends upward (convex from below). Overall average from 0 to 3 hours: 60 ÷ 3 = 20 L/h, which sits between the early slow rate and the later fast rate. Do not confuse the overall average with the instantaneous story the curve tells.
Verbal cue list:
- “increased at a constant rate” → straight line segment
- “rate of increase grew” / “accelerated” → steepening curve
- “levelled off” → flattening toward horizontal
- “decreased steadily” → straight downward slope
Combining absolute, relative, and rate in one stem
A faculty’s NSFAS-supported headcount rises from 2 000 to 2 300 over 2 years.
- Absolute change = 300 students
- Relative change = 300 ÷ 2 000 = 0.15 = 15%
- Average rate of increase = 300 ÷ 2 = 150 students per year
All three answers can appear as options; match the question’s wording (by how many, by what percent, per year) before calculating.
Common traps
| Trap | Wrong move | Correct move |
|---|---|---|
| Percent change | Divide by the new value | Divide by the original unless told otherwise |
| Average speed | Average two speeds with (v₁ + v₂)/2 when times differ | Prefer total distance ÷ total time |
| Steepness | Compare only final heights | Compare rise/run on each interval |
| Curved graph | Quote one slope for the whole curve | Report interval rates or say the rate is changing |
| Units | Mix minutes with hours in one division | Convert to consistent units first |
Practice habit
- Underline whether the stem wants absolute, percent, or rate.
- Write (new − old) and, if needed, ÷ old.
- For graphs, pick two clear points and compute rise ÷ run.
- If successive interval rates differ, say the rate is changing — that is what curvature encodes.
Shape and space (next section) reuse the same careful unit habits when perimeter, area, and volume enter the story.
A residence meal plan rises from R3 600 to R4 140. What is the relative increase?
A campus shuttle travels 150 km in 2.5 hours. What is its average speed?
On a score-versus-hours graph, scores rise from 30 to 42 between 1 and 4 hours of study. What is the average rate of score increase on that interval?
Cumulative rainwater (litres) after 0, 1, and 2 hours is 0, 8, and 24. Which statement is correct?