Gradients as rates

On any straight-line graph the gradient is change in the vertical quantity ÷ change in the horizontal quantity. Its units are "vertical unit per horizontal unit": litres per minute, pounds per mile, centimetres per hour.

1. a tank holds 20 litres at 0 minutes and 80 litres at 12 minutes
2. change in volume = 80 − 20 = 60 litres
3. change in time = 12 minutes
4. gradient = 60 ÷ 12 = 5 litres per minute

Use the change in the vertical quantity. Dividing 80 by 12 forgets the 20 litres that were there at the start.

Where the line crosses the vertical axis, the horizontal quantity is 0. So the intercept is the value before anything happens: a taxi's fixed charge, a plumber's call-out fee, a candle's height before it is lit.

On a falling line (a bath draining, fuel being used) the gradient is negative. In words, give the rate as an amount used or lost each unit: "6 litres per 100 km", not "−0.06".

You do not need a graph. Two readings at a constant rate are enough:

1. at 09:50 a tank holds 300 litres; at 10:15 it holds 175 litres
2. 09:50 to 10:15 is 25 minutes (not 10.15 − 9.50 = 0.65)
3. 300 − 175 = 125 litres in 25 minutes
4. rate = 125 ÷ 25 = 5 litres per minute

When two things work together, add the rates, then divide the job by the total rate. Two taps at 6 and 9 litres per minute fill a 300-litre tank in 300 ÷ 15 = 20 minutes.

Set a line through (0, 20) and (12, 80). Read its gradient and intercept, then imagine the axes are time in minutes and volume in litres: what does each number mean?