Cambridge IGCSE Mathematics 0580: Rates, time and money – Study Guide

This study guide teaches three subtopics of Topic 1 Number in the Cambridge IGCSE Mathematics 0580 syllabus for exams in 2025, 2026 and 2027 (version 3): 1.12 Rates, 1.15 Time and 1.16 Money. The learning outcomes are the same in the Core column (C1.12, C1.15, C1.16) and the Extended column (E1.12, E1.15, E1.16), so nothing here is Extended only. Core candidates sit Paper 1 (non-calculator) and Paper 3 (calculator); Extended candidates sit Paper 2 (non-calculator) and Paper 4 (calculator). The worked examples use numbers you can handle by hand.

The topic-level Number study guide and Number revision notes give the short version; this guide goes deeper with new examples. For the whole course, see the Cambridge IGCSE Mathematics hub and the printable checklist.

What this unit covers

Section	What you must be able to do	Tier
1.12 Rates (1)	Use common measures of rate: hourly pay, exchange rates, flow rates, fuel consumption	Both
1.12 Rates (2)	Apply other measures of rate: pressure, density, population density (formulas given)	Both
1.12 Rates (3)	Solve average speed problems (you must know the formula)	Both
1.15 Time (1)	Calculate with s, min, h, days, weeks, months, years (1 year = 365 days)	Both
1.15 Time (2)	Use the 24-hour and 12-hour clock	Both
1.15 Time (3)	Read clocks and timetables, including time zones and time differences	Both
1.16 Money (1)	Calculate with money	Both
1.16 Money (2)	Convert from one currency to another	Both

1.12 Rates

A rate compares two different quantities: dollars per hour, litres per minute, kilometres per hour. “Per” means “divided by”, so km/h is kilometres ÷ hours.

Common measures of rate

Hourly rates of pay. Pay = hourly rate × hours worked. Overtime at “time and a half” is paid at 1.5 × the normal rate.

Worked example 1 (non-calculator). Amira is paid £14.60 per hour for a 38-hour week, plus 6 hours of overtime at time and a half. Calculate her pay for the week.

basic pay      = 38 x 14.60      = 554.80
overtime rate  = 1.5 x 14.60     = 21.90 per hour
overtime pay   = 6 x 21.90       = 131.40
total          = 554.80 + 131.40 = 686.20, so £686.20

Flow rates. A flow rate is a volume per unit of time. Time = volume ÷ flow rate.

Worked example 2. A tank holding 1.8 m³ is filled from empty at 24 litres per minute. How long does it take?

1 m3 = 1000 litres, so 1.8 m3 = 1800 litres
time = 1800 / 24 = 75 minutes = 1 hour 15 minutes

Put the volume and the flow rate in the same unit before dividing.

Fuel consumption is given either as litres per 100 km or as kilometres per litre. Read which one the question uses.

Worked example 3. A car uses 6.4 litres of fuel per 100 km. Fuel costs £1.85 per litre.

(a) Fuel for a 350 km journey: 350 ÷ 100 = 3.5, and 3.5 × 6.4 = 22.4 litres.

(b) Cost: 22.4 × 1.85 = £41.44.

(c) In km per litre: 100 ÷ 6.4 = 15.625 km per litre.

Exchange rates are rates too; they are taught under 1.16 Money below.

Other measures of rate

The syllabus says the formulas for pressure, density and population density will be given in the question. You still need to substitute correctly and match the units.

density            = mass / volume           e.g. g/cm3
pressure           = force / area            e.g. N/m2
population density = population / area       e.g. people per km2

Worked example 4 (calculator).

(a) A metal block has mass 540 g and volume 200 cm³. Density = 540 ÷ 200 = 2.7 g/cm³.

(b) Volume of 1 kg of the same metal: volume = mass ÷ density = 1000 ÷ 2.7 = 370.37… = 370 cm³ (3 s.f.). The mass must be in grams to match g/cm³.

(c) A force of 450 N acts on 0.25 m². Pressure = 450 ÷ 0.25 = 1800 N/m².

(d) A town of 84 000 people covers 35 km². Population density = 84 000 ÷ 35 = 2400 people per km².

Average speed

You must know this formula; it is not given.

average speed = total distance / total time
distance = speed x time          time = distance / speed

The time must be in the unit the speed uses. Minutes become hours by dividing by 60: 36 minutes = 0.6 hours.

Worked example 5 (non-calculator). A train travels 156 km in 1 hour 36 minutes. Find its average speed.

1 h 36 min = 1 + 36/60 = 1.6 hours
speed = 156 / 1.6 = 1560 / 16 = 97.5 km/h

Multiplying both numbers by 10 removes the decimal divisor, which is the usual trick on the non-calculator papers.

Worked example 6, a journey in stages. A driver travels 90 km at 60 km/h, stops for 20 minutes, then travels 50 km in 40 minutes. Find the average speed for the whole journey.

stage 1 time   = 90 / 60 = 1.5 hours
total time     = 1.5 h + 20 min + 40 min = 2.5 hours
total distance = 90 + 50 = 140 km
average speed  = 140 / 2.5 = 56 km/h

The stop counts in the total time. Averaging the two speeds (60 and 75 km/h) is wrong.

Changing units of speed. 1 km = 1000 m and 1 hour = 3600 seconds:

54 km/h = 54 000 m / 3600 s = 15 m/s
12 m/s  = 12 x 3600 = 43 200 m per hour = 43.2 km/h

At 15 m/s for 2 minutes, a cyclist covers 15 × 120 = 1800 m. Change the 2 minutes to 120 seconds first, because the speed is per second.

1.15 Time

Units of time

60 s = 1 min    60 min = 1 h    24 h = 1 day    7 days = 1 week
12 months = 1 year    1 year = 365 days

Time is not decimal: there are 60 minutes in an hour, not 100.

Worked example 7.

(a) Seconds in one week: 7 × 24 × 60 × 60 = 604 800 seconds.

(b) Write 2000 minutes in days, hours and minutes. 2000 ÷ 60 = 33 remainder 20, so 33 h 20 min, and 33 h = 1 day 9 h. Answer: 1 day 9 hours 20 minutes.

(c) A calculation gives 2.35 hours. 0.35 × 60 = 21 minutes, so this is 2 hours 21 minutes, not 2 hours 35 minutes.

(d) 18 months = 1.5 years, and 2 years = 2 × 365 = 730 days.

The 24-hour and 12-hour clock

The syllabus writes 24-hour times with four digits, such as 03 15 for 3.15 a.m. and 15 15 for 3.15 p.m. From 1 p.m. onwards, add 12 to the hour. Midnight is 00 00 and noon is 12 00.

Reading a clock face. The short hand shows the hour, the long hand the minutes; each small mark is 1 minute. Long hand on 9, short hand between 3 and 4 in the afternoon: 3.45 p.m., or 15 45.

11.05 p.m.  -> 23 05         00 45 -> 12.45 a.m.
12.05 p.m.  -> 12 05         19 30 -> 7.30 p.m.

Intervals across midnight. Count on to the next hour, then whole hours, then the remaining minutes.

Worked example 8. A night shift runs from 22 38 to 06 15 the next morning. How long is it?

22 38 -> 23 00    22 min
23 00 -> 06 00     7 h
06 00 -> 06 15    15 min
total              7 h 37 min

Reading timetables

Each column is one journey and each row is one stop.

Harbour      06 50   07 35   08 20   09 05
Old Town     07 12   07 57   08 42   09 27
University   07 31   08 16   09 01   09 46
Park Gate    07 55   08 40   09 25   10 10

Worked example 9. Use this tram timetable.

(a) Harbour to Park Gate: 06 50 to 07 55 is 1 hour 5 minutes.

(b) Leo boards at Old Town and must be at Park Gate by 09 00. The 08 42 from Old Town arrives at 09 25, too late, so the latest tram he can catch is the 07 57, which arrives at 08 40.

(c) Mia reaches University at 08 05. The next tram leaves at 08 16, so she waits 11 minutes.

Time zones and local time

If a city is ahead, its clocks show a later time: add the difference. If it is behind, subtract. Work the journey in one city’s time, then convert once at the end.

Worked example 10. City Y is 3 hours behind City X. A flight leaves X at 01 20 local time and lasts 9 h 50 min. Find the local time in Y when it lands.

in X time:  01 20 + 9 h 50 min = 11 10
Y is 3 h behind:  11 10 - 3 h = 08 10, local time in Y

Worked example 11. A flight leaves Y at 22 15 local time and lands in X at 11 40 the next day, local time in X. How long is the flight?

landing in Y time:  11 40 - 3 h = 08 40 the next day
22 15 -> 08 40:     45 min + 9 h + 40 min = 10 h 25 min

Both times must be in the same zone before you find the difference.

1.16 Money

Calculating with money

Give money to 2 decimal places: £26.3 is written £26.30. Decide whether to round up or down in context.

Worked example 12 (non-calculator). A phone plan costs €18.50 a month including 300 minutes of calls. Extra minutes cost €0.07 each. Rafael uses 412 minutes. Find his bill.

extra minutes = 412 - 300 = 112
extra cost    = 112 x 0.07 = 784 / 100 = 7.84
bill          = 18.50 + 7.84 = 26.34, so €26.34

Worked example 13. Six friends share a bill of £187.40 equally. 187.40 ÷ 6 = 31.2333… Each must pay £31.24 to cover the bill; at £31.23 each they would pay only 6 × 31.23 = £187.38.

Converting between currencies

An exchange rate such as €1 = $1.10 works like any other rate:

• euros to dollars: multiply by 1.10, so €250 = 250 × 1.10 = 275 dollars;
• dollars to euros: divide by 1.10, so 440 dollars = 440 ÷ 1.10 = €400.

Check the size of the answer. Each euro is worth more than one dollar, so the number of dollars should be larger.

Worked example 14, two rates. On one day, €1 = $1.10 and €1 = ¥160 (Japanese yen).

(a) Change 440 dollars into yen. Go through euros: 440 dollars = €400, then 400 × 160 = ¥64 000.

(b) A watch costs ¥27 200 in Tokyo and $185 in New York. Where is it cheaper, and by how much in dollars?

27 200 / 160 = 170 euros
170 x 1.10   = 187 dollars
187 - 185    = 2, so it is cheaper in New York by 2 dollars

Convert both prices into the same currency before comparing.

Common errors

• Writing 1 hour 36 minutes as 1.36 hours instead of 1.6 hours, or reading 2.35 hours as 2 h 35 min.
• Averaging two speeds, or leaving a stop out of the total time.
• Converting only half of a compound unit, such as km to m but not h to s.
• Using kilograms with a density in g/cm³.
• Adding a time difference when the city is behind.
• Multiplying instead of dividing in a currency conversion. Check which currency should give the bigger number.
• Comparing prices in two currencies without changing them to one currency.
• Writing £7.8 instead of £7.80, or rounding down when a shared cost must be covered.

Next steps

• Practise with Number practice questions and, for Extended, Number (Extended) practice questions.
• Find your gaps with the free Core diagnostic or Extended diagnostic.
• Find other resources for syllabus 0580 on the course hub.

Official syllabus

Cambridge IGCSE Mathematics 0580 syllabus for exams in 2025, 2026 and 2027, version 3 (published May 2024), Cambridge University Press & Assessment. Subtopics C1.12/E1.12 Rates, C1.15/E1.15 Time and C1.16/E1.16 Money.