CourseCourseQ BankQuestion BankDocsDocuments

C1.12 Rates

Syllabus
0580–2028–2029
Topic
C1.12
Level
Core

Learning objectives

C1.12.1Use common measures of rate.• Use common measures of rate.C1.12.2Apply other measures of rate.• Apply other measures of rate.C1.12.3Solve 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).

Read and calculate with common rates

A rate compares quantities with different units. The unit tells you both the division and the calculation direction: 15.20perhourmeans15.20 per hour means$15.20$ for each hour, while 20 litres per minute means 20 litres for each minute.

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

Context Read the units Typical calculation
Hourly pay dollars/hour pay == hours ×\times hourly rate
Exchange rate new currency/old currency old amount ×\times rate == new amount
Flow rate litres/minute volume == flow rate ×\times time
Fuel use litres/100 km or km/litre follow the stated unit; the two forms are not interchangeable

At \15.20perhour,40hoursgivesper hour, 40 hours gives40\times15.20=$608.If1dollarexchangesfor1.25euros,. If 1 dollar exchanges for 1.25 euros,80\times1.25=€100$; converting back divides by 1.25. Write the units beside the numbers so the unwanted unit cancels.

Convert the reference unit before applying the rate. For example, 8484 km/h is 84 000÷60=140084\,000\div60=1400 m/min. Multiplying by 1000 changes kilometres to metres; dividing by 60 changes 'per hour' to 'per minute'.

Do not multiply automatically. Use the rate's written units to decide whether to multiply or divide, and do not assume that every fuel-consumption figure is expressed in the same direction.

Use a given formula for pressure and density rates

Pressure, material density and population density are rates: one quantity is measured per unit of another. In this objective, the required formula is supplied in the question; your job is to match each value and unit to that formula.

Measure Common given formula Meaning of the unit
Pressure P=F/AP=F/A force per unit area, such as N/m²
Density ρ=m/V\rho=m/V mass per unit volume, such as g/cm³
Population density d=N/Ad=N/A people per unit area, such as people/km²
  1. Copy the formula given in the question. 2. Convert values into compatible units. 3. Substitute with units. 4. Rearrange only if the unknown is not already isolated. 5. State the compound unit in the same order as the formula.

A metal cuboid has mass 4 kg and volume 600 cm³. Convert 44 kg to 40004000 g, then ρ=4000÷600=6.67\rho=4000\div600=6.67 g/cm³ (3 s.f.). If density and volume are known, rearrange ρ=m/V\rho=m/V to m=ρVm=\rho V.

For a population of 735 000 across 1 477 300 km², population density is 735 000÷1 477 300≈0.498735\,000\div1\,477\,300\approx0.498 people/km². A value below 1 is possible: it means fewer than one person per square kilometre on average, not a fraction of a person at each location.

Do not memorise extra physics formulas as syllabus requirements here. Use the formula supplied, keep numerator and denominator in the correct order, and convert mass, area or volume units before dividing.

Find average speed from total distance and total time

Average speed describes a whole journey. It is the total distance travelled divided by the total elapsed time, including any stops when the question treats them as part of the 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}

  1. Add every distance that belongs to the journey. 2. Add every travel time and included stop time. 3. Convert all time to the unit required by the speed. 4. Divide total distance by total time. 5. Attach the compound unit, such as km/h or m/s.

A cyclist travels 45 km in 3 hours 45 minutes. Since 45 minutes is 45/60=0.7545/60=0.75 hours, the total time is 3.75 hours. The average speed is 45÷3.75=1245\div3.75=12 km/h.

For a two-stage run, first find missing totals. Running for 45 minutes at 9.5 km/h covers 9.5×0.75=7.1259.5\times0.75=7.125 km. Running 8.1 km at 7.5 km/h takes 8.1÷7.5=1.088.1\div7.5=1.08 h. So the whole-run average is (7.125+8.1)÷(0.75+1.08)≈8.32(7.125+8.1)\div(0.75+1.08)\approx8.32 km/h.

To convert m/s to km/h, multiply by 60×60÷1000=3.660\times60\div1000=3.6. To convert km/h to m/s, divide by 3.6. A decimal hour is not decimal minutes: 2.4 hours is 2 hours 24 minutes.

Do not average two speeds unless their travel times are equal. Average speed always uses total distance divided by total time; it is not generally the arithmetic mean of the stage speeds.