How Fast Is It Changing? Rates of Change

Imagine a tap drips steadily into a bucket. After 2 hours the bucket holds 6 litres. After 5 hours it holds 15 litres. Is the drip speeding up, slowing down, or staying the same? How would you even begin to answer that? Today's lesson gives you the mathematical tool to decide — and to predict what happens next.

A rate of change tells you how much one quantity changes for every unit change in another quantity — usually time. The formula is: Rate of change = (change in quantity) ÷ (change in time) In maths notation we often write this as: rate = Δy ÷ Δx The symbol Δ (the Greek letter delta) means 'the change in'. So Δy means 'how much y changed' and Δx means 'how much x changed'. If the rate of change is constant (the same throughout), the relationship is linear — it forms a straight line on a graph. If the rate changes at different points, the relationship is non-linear. Units matter: always pair the unit of the quantity with the unit of time, for example litres per hour, metres per second, or pounds per day. WORKED EXAMPLE: The table below shows water volume in a tank at two moments. Time (hours) | Volume (litres) 0 | 10 4 | 30 Step 1 — Find the change in volume (Δy): 30 − 10 = 20 litres Step 2 — Find the change in time (Δx): 4 − 0 = 4 hours Step 3 — Divide: Rate = 20 ÷ 4 = 5 litres per hour Interpretation: The tank is filling at 5 litres every hour. Prediction: After 6 hours the volume will be 10 + (5 × 6) = 40 litres.