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Cambridge IGCSE0580

Rates

Mathematics 0580 Chapter Notes

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Rates
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In this chapter

  1. 1Pay and exchange rates
  2. 2Flow rates and fuel consumption
  3. 3Density and population density
  4. 4Speed, distance and time
  5. 5Finding an arrival time
  6. 6Average speed for a whole journey
  7. 7Changing the units of a rate

1. Pay and exchange rates

A rate compares two quantities, such as money per hour or one currency per unit of another. An hourly rate of pay is the money earned for each hour worked.

An exchange rate says how much of one currency you get for one unit of another. Work out first whether the answer should be a bigger or a smaller number. Then multiply or divide to match.

pay = hourly rate × number of hours

dollars ÷ dollars per euro = euros

Examiner insight

“Common errors included finding the increase in the hourly pay of $3.60 and then forgetting to add it on to the basic pay.” “There was a minority who multiplied rather than divided by the exchange rate.”

Worked example 12 marks

Do not use a calculator. The exchange rate is 1 euro = 1.25 dollars. Maya changes 400 dollars into euros. How many euros does she receive?

  1. 1

    One euro is worth more than one dollar, so the answer is smaller than 400. Divide: 400 ÷ 1.25 [1]

  2. 2

    400 ÷ 1.25 = 320 euros [1]

2. Flow rates and fuel consumption

A flow rate is the volume that flows in each unit of time, such as millilitres per second. Use the rate to find the time for a given volume, or the volume for a given time.

Fuel consumption is also a rate: fuel used for a distance, or distance for an amount of fuel. Keep the units consistent, and change litres and millilitres before you divide.

time = volume/(flow rate)

1 litre = 1000 ml

Examiner insight

“Common errors included: using the diameter rather than the radius of the pipe when calculating the cross-sectional area, using inconsistent units in the calculation, or using an incorrect conversion factor between cubic centimetres and litres.”

Worked example 13 marks

A tank contains 268 litres of water. The water from the tank is emptied into a cylindrical container at a rate of 300 ml per second. Calculate the time taken for the tank to be completely emptied. Give your answer in minutes and seconds, correct to the nearest second.

  1. 1

    268 litres = 268 × 1000 = 268 000 ml [1]

  2. 2

    time = 268 000 ÷ 300 = 893.3… s [1]

  3. 3

    893.3… s = 14 minutes 53 seconds [1]

3. Density and population density

Density is mass per unit volume. Population density is the number of people per unit area. Pressure is another measure of rate.

The syllabus says the required formulas will be given in the question. Read the formula, then put in the numbers with their units.

density = mass/volume

population density = population/area

Examiner insight

“Incorrect conversion of units was the most common error with the volume divided by a variety of powers of 10 to convert from cubic centimetres to cubic metres.”

Worked example 13 marks

The table gives some information about two towns, A and B. The population density of the two towns is the same. Calculate the population of town B. Population density = population ÷ area.

populationarea (km²)
town A4680104
town B760
  1. 1

    density of A = 4680 ÷ 104 = 45 people per km² [1]

  2. 2

    population of B = 45 × 760 [1]

  3. 3

    = 34 200 [1]

4. Speed, distance and time

Knowledge of the speed, distance, time formula is required. Divide distance by time to find speed.

The units must match. For km/h, the time must be in hours. Divide minutes by 60, not by 100, so 1 hour 15 minutes is 1.25 hours.

speed = distance/time

distance = speed × time

time = distance/speed

Examiner insight

“Nearly all candidates knew that they must divide the distance by the time but expressing the time in a suitable form was a problem for many.”

Worked example 12 marks

A bus takes 1 hour 15 minutes to travel 45 km between two towns. Work out the average speed of the bus in kilometres per hour.

  1. 1

    time = 1 + 15/60 = 1.25 hours, so speed = 45 ÷ 1.25 [1]

  2. 2

    = 36 km/h [1]

5. Finding an arrival time

To find an arrival time, find the time taken first. Then change the decimal part of the hours into minutes by multiplying by 60. Add the result to the start time.

Do not round the hours before converting. Use a fraction or the full calculator value.

2.65 h = 2 h + 0.65 × 60 min = 2 h 39 min

Examiner insight

“Errors then followed when attempting to change this time to hours and minutes, commonly because of premature rounding.” “Common errors included the incorrect conversion to 2 hours 40 minutes or 2 hours 4 minutes; giving 2.4 hours or their equivalent as the final answer; and giving the incomplete 4.54 as the final answer.”

Worked example 13 marks

Sam leaves home at 14 30 and drives 106 km to town A. His average speed is 40 km/h. Calculate the time he arrives at town A.

  1. 1

    time = 106 ÷ 40 = 2.65 hours [1]

  2. 2

    0.65 × 60 = 39 minutes, so the journey takes 2 hours 39 minutes [1]

  3. 3

    14 30 + 2 h 39 min = 17 09 [1]

6. Average speed for a whole journey

For a journey in stages, find the total distance and the total time. Then divide one by the other.

Do not find the mean of the separate speeds. The stages take different times, so the speeds do not count equally.

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

Examiner insight

“Common errors included the use of 8 km rather than 16, the use of 24 minutes rather than 58, incorrect or no conversion to hours, and finding the speeds of the separate parts of the journey and then attempting to combine these answers into an average figure.”

Worked example 14 marks

Ali drives 56 km at an average speed of 48 km/h. He then drives for 30 minutes at 54 km/h. Calculate the average speed for his whole journey.

  1. 1

    first stage: time = 56 ÷ 48 = 7/6 hours [1]

  2. 2

    second stage: distance = 54 × 0.5 = 27 km [1]

  3. 3

    total distance = 56 + 27 = 83 km and total time = 7/6 + 1/2 = 5/3 hours [1]

  4. 4

    average speed = 83 ÷ 5/3 = 49.8 km/h [1]

7. Changing the units of a rate

A rate has two units, so change each one separately. For km/h to m/s, change kilometres to metres and hours to seconds.

Multiply by 1000 for the distance and divide by 3600 for the time. Write the units at every step to see which way to go.

1 km/h = 1000/3600 m/s

1 h = 60 × 60 = 3600 s

Examiner insight

“This part on speed conversion was found challenging, although many candidates were able to convert to metres per hour by multiplying by 1000.” “Common errors included multiplying by 10 or 100, dividing by 60 and multiplying by either 60 or 3600.”

Worked example 12 marks

Llew cycles to school at an average speed of 18.9 kilometres per hour. Write this speed in metres per second.

  1. 1

    18.9 km = 18.9 × 1000 = 18 900 m and 1 hour = 3600 s [1]

  2. 2

    18 900 ÷ 3600 = 5.25 m/s [1]

Recap

  • A rate compares two quantities: pay per hour, volume per second, mass per volume, people per area.
  • Exchange rates: decide if the answer should be bigger or smaller, then multiply or divide.
  • Flow rate: time = volume ÷ flow rate. Change litres and millilitres first.
  • Speed = distance ÷ time. Time in hours means minutes ÷ 60.
  • Average speed = total distance ÷ total time. Never average the speeds.
  • km/h to m/s: multiply by 1000 and divide by 3600.

End-of-chapter exercise

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

  1. Do not use a calculator. Priya is paid $9.50 per hour. She works for 8 hours. Work out her pay.2 marks
  2. Do not use a calculator. The exchange rate is 1 euro = 1.20 dollars. A hotel room costs 300 dollars. Work out the cost in euros.2 marks
  3. Do not use a calculator. A cyclist travels 45 km in 3 hours 45 minutes. Work out the average speed in km/h.2 marks
  4. A pump moves water at 250 ml per second. How many minutes does it take to fill a 30 litre tank?3 marks
  5. Do not use a calculator. A metal block has mass 600 g and volume 80 cm³. Density = mass ÷ volume. Work out the density of the metal in g/cm³.2 marks
  6. A region has area 8.5 km². Its population density is 240 people per km². Work out the population of the region.2 marks
  7. Do not use a calculator. A train leaves at 09 40 and travels 120 km at an average speed of 90 km/h. Work out the time it arrives.3 marks
  8. Do not use a calculator. Change 72 km/h into m/s.2 marks
  9. A ferry sails 18 km to an island at an average speed of 24 km/h. It waits at the island for 12 minutes. It then sails a further 21 km at an average speed of 7.5 m/s. Calculate the average speed of the ferry for the whole journey, including the wait. Give your answer in km/h, correct to 3 significant figures.5 marks
  10. A lorry travels 315 km at an average speed of 54 km/h. It uses 28 litres of fuel for every 100 km. Fuel costs 1.35 euros per litre. The driver is paid 16.80 dollars per hour. The exchange rate is 1 euro = 1.12 dollars. Calculate the total cost, in dollars, of the fuel and the driver's pay for the journey. Give your answer correct to the nearest cent.6 marks

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