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E1.12 Rates

Syllabus
0580–2028–2029
Topic
E1.12
Level
Extended

Learning objectives

E1.12.1Use common measures of rate.• Use common measures of rate.E1.12.2Apply other measures of rate.• Apply other measures of rate.E1.12.3Solve problems involving average speed• Solve problems involving average speed. e.g. calculate with: • hourly rates of pay • exchange rates between currencies • flow rates • fuel consumption. e.g. calculate with: • pressure • density • population density. Required formulas will be given in the question. Knowledge of speed/distance/time formula is required. e.g. A cyclist travels 45 km in 3 hours 45 minutes. What is their average speed? Notation used will be, e.g. m/s (metres per second), g/cm³ (grams per cubic centimetre).

Let rate units choose the operation

A rate compares quantities with different units. Read 'per' as division: 18 litres per minute means 18 litres for each minute. The written unit shows which quantity is produced and which reference quantity it is measured against.

rate=quantityreference quantityquantity=rate×reference quantity\text{rate}=\frac{\text{quantity}}{\text{reference quantity}}\qquad \text{quantity}=\text{rate}\times\text{reference quantity}

Context Rate unit Use
hourly pay dollars/hour pay == rate ×\times hours
currency exchange new currency/old currency multiply in the stated direction; divide to reverse it
flow litres/minute volume == rate ×\times time
fuel use litres/100 km fuel per 100 km =total fueltotal distance×100=\dfrac{\text{total fuel}}{\text{total distance}}\times100

A machine makes 11 components every 15 minutes. In 6 hours there are 6×60÷15=246\times60\div15=24 intervals, so it makes 24×11=26424\times11=264 components. Converting the time first keeps the rate's reference unit consistent.

Units also guide conversion. A flow of 3.6 m3^3/h is 36003600 litres per hour, then 3600÷60=603600\div60=60 litres per minute. Apply a conversion to the quantity named by that part of the compound unit.

Do not multiply automatically: divide when finding a rate from two totals, and multiply when a rate and reference quantity are known. Fuel consumption in litres/100 km is not the same quantity as efficiency in km/litre.

Apply a supplied rate formula with compatible units

Pressure, material density and population density each measure one quantity per unit of another. The required formula is supplied in the question; identify its numerator and denominator, then make their units compatible before substituting.

Measure Typical supplied formula Meaning of result
pressure P=F/AP=F/A force per unit area, such as N/m2^2
density ρ=m/V\rho=m/V mass per unit volume, such as g/cm3^3
population density d=N/Ad=N/A people per unit area, such as people/km2^2

Copy the given formula, convert the data to the units required by the answer, substitute, and rearrange only if the unknown is not isolated. Preserve the numerator/denominator order in the compound unit.

A cuboid of sand measures 0.600.60 m by 0.300.30 m by 0.250.25 m and has mass 54 kg. Its volume is 0.60×0.30×0.25=0.0450.60\times0.30\times0.25=0.045 m3^3, so ρ=54÷0.045=1200\rho=54\div0.045=1200 kg/m3^3.

If density and volume are known, m=ρVm=\rho V; if density and mass are known, V=m/ρV=m/\rho. Area and volume conversions must be squared or cubed: 11 m2=10000^2=10000 cm2^2 and 11 m3=1000000^3=1000000 cm3^3.

A density below 1 person/km2^2 is meaningful as an average over an area. Do not interpret it as a fractional person at one location, and do not mix kilograms with cubic centimetres unless the final unit deliberately combines them.

Average speed uses whole-journey totals

Average speed describes an entire journey: divide total distance travelled by total elapsed time. Include every journey section and any stop whose time is included in the stated journey.

average speed=total distancetotal timed=stt=ds\text{average speed}=\frac{\text{total distance}}{\text{total time}}\qquad d=st\qquad t=\frac{d}{s}

Add the relevant distances, add the relevant times, convert time to the unit required by the speed, then divide. For 72 km in 1 h 36 min, 36/60=0.636/60=0.6 h, so the average speed is 72÷1.6=4572\div1.6=45 km/h.

For 30 km at 60 km/h followed by 40 km at 40 km/h, the times are 30/60=0.530/60=0.5 h and 40/40=140/40=1 h. The whole-journey average is (30+40)÷(0.5+1)=46.7(30+40)\div(0.5+1)=46.7 km/h, not the mean of 60 and 40.

To convert m/s to km/h, multiply by 3600/1000=3.63600/1000=3.6; reverse the conversion by dividing by 3.6. A decimal hour must be converted by multiplying its fractional part by 60: 2.4 h is 2 h 24 min.

Never average stage speeds unless the stage times are equal. Average speed is total distance divided by total time, and its unit must match the distance and time units used in that division.