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Cambridge IGCSE0607

Rates

International Mathematics 0607 Chapter Notes

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Rates
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In this chapter

  1. 1Common measures of rate
  2. 2Speed, distance and time
  3. 3Time in hours and minutes
  4. 4Average speed for a whole journey
  5. 5Changing the units of a speed

1. Common measures of rate

A rate tells you how much of one quantity there is for each one unit of another. The word "per" shows it: pay per hour, litres per minute, litres of fuel per 100 km.

The syllabus examples are an hourly rate of pay, an exchange rate between currencies, a flow rate and fuel consumption. Each one is solved the same way: find what one unit gives, then scale up or down.

Some costs are a fixed charge plus a charge per kilometre. The fixed charge is paid once. Only the charge per kilometre is multiplied by the distance.

total pay = rate per hour × hours worked

time to fill = volume/(flow rate)

Examiner insight

“Most candidates recognised the need to divide the capacity by the rate of flow but the conversion of units, particularly the litres, proved too difficult for all but the best.” “Many candidates gained full marks and the common difficulty in interpretation was in distinguishing between a fixed charge and a charge per kilometre.”

Worked example 12 marks

Do not use a calculator. A car uses 6 litres of fuel for every 100 km. How many litres does it use for a journey of 250 km?

  1. 1

    250 ÷ 100 = 2.5 lots of 100 km [1]

  2. 2

    2.5 × 6 = 15 litres [1]

2. Speed, distance and time

Speed is the distance travelled per unit time. The three quantities are linked by one formula, which you rearrange to find whichever is missing.

Use matching units. Distance in km and time in hours give speed in km/h. Distance in m and time in s give speed in m/s.

Write km/h in full. Do not write kph or km/hr.

speed = distance/time

distance = speed × time

time = distance/speed

Examiner insight

“Candidates should know to use international units, that is, km/h and not kph nor km/hr.” “Many incorrectly multiplied the distance in km by a time in minutes rather than in hours.”

Worked example 12 marks

A car travels at a speed of 90 km / h for 1 hour 20 min. Work out the distance the car travels.

  1. 1

    20 min = 1/3 h, so time = 11/3 = 4/3 h [1]

  2. 2

    distance = 90 × 4/3 = 120 km [1]

3. Time in hours and minutes

There are 60 minutes in an hour, not 100. To change minutes to hours, divide by 60: 45 min is 3/4 h.

To change a decimal time back, multiply the decimal part by 60. So 1.6 h is 1 h and 0.6 × 60 = 36 min.

Without a calculator, keep the time as a fraction. With one, do not round the time until the final answer.

minutes ÷ 60 = hours

decimal part of an hour × 60 = minutes

Examiner insight

“The conversion of 2 hours 45 minutes into 2.75 hours was usually done although 2.45 hours was often seen.” “A small number of candidates, having found the time of 1.6 hours, converted this to 1 hour 6 minutes.”

Worked example 13 marks

Lavinia leaves home at 18 25 to go to the cinema. The cinema is 750 metres from Lavinia's home. Lavinia walks at an average speed of 5 km/h. Calculate the time Lavinia arrives at the cinema.

  1. 1

    750 m = 0.75 km, so time = 0.75 ÷ 5 = 0.15 h [1]

  2. 2

    0.15 × 60 = 9 minutes [1]

  3. 3

    18 25 + 9 min = 18 34 [1]

4. Average speed for a whole journey

Average speed is total distance divided by total time. It is not the mean of the speeds.

For a journey in stages, find the time of each stage. Add the distances, add the times, then divide.

If the journey includes a stop, the stopping time is part of the total time.

average speed = (total distance)/(total time)

Examiner insight

“Some candidates found the average of the 2 speeds and gave an answer of 62.5 km / h.” “The stronger candidates succeeded whilst many did not include the 30 minutes of stoppage time.”

Worked example 13 marks

A car travels 120 km at an average speed of 75 km/h. The car then travels a further 120 km at an average speed of 50 km/h. Calculate the average speed for the complete journey.

  1. 1

    first stage time = 120 ÷ 75 = 1.6 h, second stage time = 120 ÷ 50 = 2.4 h [1]

  2. 2

    total distance = 120 + 120 = 240 km, total time = 1.6 + 2.4 = 4 h [1]

  3. 3

    average speed = 240 ÷ 4 = 60 km/h [1]

5. Changing the units of a speed

There are 1000 m in a km and 3600 s in an hour. To change km/h to m/s, multiply by 1000 and divide by 3600. To change m/s to km/h, do the reverse.

For example, a speed of 20 m/s is 20 × 3600 ÷ 1000 = 72 km/h.

Look at the answer and ask if it is sensible. A person does not walk at 3000 km/h.

km/h → m/s: × 1000/3600

m/s → km/h: × 3600/1000

Examiner insight

“Errors included using 60 rather than 3600 seconds in an hour or 100 rather than 1000 metres in a kilometre or dividing by 3600 and multiplying by 1000.” “In questions like these, candidates should look at their answers to check they make sense in context – 3000 km/h is far too fast a speed and 0.5 km/h is far too slow.”

Worked example 13 marks

A cyclist rides 400 m in 25 s. Work out her average speed in km / h.

  1. 1

    speed = 400 ÷ 25 = 16 m/s [1]

  2. 2

    in one hour: 16 × 3600 = 57 600 m [1]

  3. 3

    57 600 ÷ 1000 = 57.6 km/h [1]

Recap

  • A rate is an amount per one unit: pay per hour, litres per minute, fuel per 100 km.
  • Speed = distance ÷ time, distance = speed × time, time = distance ÷ speed.
  • Minutes ÷ 60 = hours. The decimal part of an hour × 60 = minutes.
  • Average speed = total distance ÷ total time, never the mean of the speeds.
  • Stopping time counts in the total time.
  • km/h to m/s: × 1000 ÷ 3600. m/s to km/h: × 3600 ÷ 1000.
  • Write km/h, not kph or km/hr.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. State the formula for average speed.1 mark
  2. Do not use a calculator. Sam is paid 12 dollars per hour. He works for 3 hours 30 minutes. Work out how much he is paid.2 marks
  3. Do not use a calculator. A hotel bill is 5400 rupees. The exchange rate is 1 euro = 90 rupees. Work out the bill in euros.2 marks
  4. Do not use a calculator. Water flows into a tank at 15 litres per minute. How long does it take to put 360 litres into the tank?2 marks
  5. Do not use a calculator. A car travels at 80 km / h for 45 minutes. Work out the distance it travels.2 marks
  6. A cyclist travels 45 km in 3 hours 45 minutes. Work out her average speed.2 marks
  7. Do not use a calculator. Change 72 km / h into m / s.2 marks
  8. Do not use a calculator. A bus travels 100 km at an average speed of 60 km / h. It then travels a further 60 km at an average speed of 45 km / h. (a) Work out the time taken for the first 100 km. Give your answer in hours and minutes. (b) Work out the average speed for the whole journey.5 marks
  9. A van hire company charges a fixed charge of 48 francs plus 0.36 francs for each kilometre driven. Priya pays 125 euros in total to hire a van. The exchange rate is 1 euro = 0.96 francs. (a) Work out how many kilometres Priya drove. (b) The van uses 7.5 litres of fuel per 100 km. Work out how many litres of fuel the van used.5 marks
  10. A delivery drone leaves its base at 10 50 and flies a route in three stages. Stage 1 is 18 km at an average speed of 45 km / h. The drone then waits for 12 minutes. Stage 2 is 24 km and takes 34 minutes. Stage 3 is 18 km back to the base. The average speed for the whole trip, including the wait, is 40 km / h. (a) Find the time the drone arrives back at its base. (b) Work out the average speed for stage 3. Give your answer in m / s.7 marks

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