Time, Currency and Conversions

Mathematics (Core) revision notes · Cambridge (CIE) · 0580 · Core

1. Units of time

Conversions to know

Time does not use the decimal system, so its conversions have to be learned rather than derived

Unit	Equals
1 minute	60 seconds
1 hour	60 minutes
1 day	24 hours
1 week	7 days
1 year	12 months, taken as 365 days

This specification treats a year as 365 days unless a question states otherwise
Month lengths are needed for date calculations: April, June, September and November have 30 days, February has 28 (29 in a leap year), and the remaining seven have 31

Minutes are not decimals of an hour

Sixty minutes make an hour, so a part-hour written as a decimal is a fraction of 60, not of 100
- 15 minutes is 0.25 h, 30 minutes is 0.5 h, and 45 minutes is 0.75 h
Writing 2 hours 30 minutes as 2.30 hours is wrong; it is 2.5 hours

Worked example

Changing a decimal time into hours and minutes

A calculation gives a journey time of 2.458 hours. Write it in hours and minutes, to the nearest minute. Another gives a time of 1427 seconds. Write it in minutes and seconds.

Solution:

The whole hours stay as they are: 2 hours
Only the decimal part is a fraction of an hour, so multiply it by 60: 0.458 × 60 = 27.48 minutes
Round at this last step: the journey takes 2 h 27 min
Rounding 2.458 to 2.46 first would give 0.46 × 60 = 27.6, which rounds to 28 minutes, one minute out
For the seconds, find the whole minutes first: 1427 ÷ 60 = 23.78…, so 23 whole minutes
Those 23 minutes are 23 × 60 = 1380 seconds, leaving 1427 − 1380 = 47 seconds
The time is 23 min 47 s, not 23 min 78 s

2. The 12-hour and 24-hour clock

Reading each format

A 12-hour clock runs twice through the day, distinguished by am and pm
- am covers midnight through to midday, and pm covers midday through to midnight
A 24-hour clock runs once through the day using four digits, from 00 00 at midnight to 23 59
Converting from pm to 24-hour form means adding 12 to the hours
- 7.40 pm becomes 19 40
The hours before 10 am keep a leading zero
- 6.05 am becomes 06 05

Three analogue clock faces read off in both formats: hands at ten past ten give 10 10 or 22 10, with five minutes between consecutive numbers; the half past one face has its hour hand halfway between 1 and 2, giving 01 30 or 13 30; and the quarter to seven face has its hour hand nearer 7 than 6, giving 06 45 or 18 45 — Source: Time by Save My Exams

The two awkward cases

Midnight is 00 00 in 24-hour form and 12 am in 12-hour form
Midday is 12 00 and 12 pm, so 12 pm is not 24 00

Exam tip

No am or pm on a 24-hour time

A 24-hour time is a time of day: it takes no am or pm, and it is not a length of time. The schemes take the final mark away from answers such as 19 40 pm, 07:15am or 19 hr 40 min. Write 19 40 or 19:40, or 7.40 pm when the question does not ask for the 24-hour clock.

3. Calculating with time

Finding a duration

Subtracting clock times directly can give a wrong answer, because an hour is 60 minutes, not 100
Count on instead: go up to the next whole hour, then in whole hours, then the remaining minutes
- From 14 25 to 18 10: 35 minutes reaches 15 00, then 3 hours reaches 18 00, then 10 minutes more, giving 3 hours 45 minutes

Counting the time from 6:32 pm to 11:04 pm in chunks: 28 minutes up to 7:00 pm, then 4 whole hours to 11:00 pm, then 4 minutes to 11:04 pm, which add to 4 hours 32 minutes — Source: Time by Save My Exams

Adding a duration works the same way, carrying 60 minutes as one hour

Exam tip

Write the length of time first

Working out a clock time from another time and a length of time was set in 8 of the 22 papers. When that length must be found first, by adding several times, changing minutes to hours or dividing a distance by a speed, the schemes give it a mark even if the clock time slips. Write it on its own line in hours and minutes, such as 200 minutes = 3 h 20 min, before adding it on.

Worked example

Adding a duration to a time

A bus leaves at 14 35 and the journey takes 2 hours 50 minutes. Work out the arrival time.

Solution:

Add the hours first: 14 35 + 2 hours = 16 35
Then add the minutes: 35 + 50 = 85 minutes
85 minutes is more than an hour, so exchange 60 of them for 1 hour, leaving 25
16 35 + 50 minutes = 17 25
The bus arrives at 17 25
Adding 2.50 to 14.35 on a calculator gives 16.85, which is not a time at all: minutes are sixtieths, not decimals

Worked example

An arrival time after midnight

A coach leaves at 19 45 and the journey takes 7 hours 25 minutes. Find the arrival time.

Solution:

Add the hours first: 19 45 + 7 hours = 26 45
Add the minutes: 45 + 25 = 70 minutes, which is 1 hour 10 minutes, so the time reaches 27 10
A day has only 24 hours, so take 24 off the hours: 27 10 is 03 10 the next day
The coach arrives at 03 10, which is 3.10 am

4. Timetables and local time

Reading a timetable

Timetable columns are journeys and rows are stopping points, so a single service is read down one column
Times are given in 24-hour form, and a blank entry means that service does not call there
A journey time is the difference between the departure and arrival entries in the same column

A bus timetable with four stops down the left and three columns of 24-hour departure times, so one column is one bus running through every stop and reading down a column follows a single journey — Source: Time by Save My Exams

Local time and time differences

Places in different time zones record different local times for the same instant
A place ahead of another has its local time later by the difference, and a place behind has it earlier
Add the difference when travelling to a place that is ahead, and subtract when it is behind
Where a question gives a flight duration, apply the duration first and the time difference second, keeping the two steps separate

5. Units of measure

The conversions to know

Metric conversions all work by multiplying or dividing by a power of ten

Measure	Conversions
Length	1 cm = 10 mm, 1 m = 100 cm, 1 km = 1000 m
Mass	1 g = 1000 mg, 1 kg = 1000 g, 1 tonne = 1000 kg
Capacity	1 litre = 1000 ml, 1 cl = 10 ml
Volume to capacity	1 ml = 1 cm³, 1 litre = 1000 cm³, 1 m³ = 1000 litres

Decide between multiplying and dividing by asking whether the count of units should go up or down
Changing to a smaller unit gives a bigger number, so multiply; changing to a larger unit gives a smaller number, so divide
Awkward conversions are safer in stages, so centimetres to kilometres goes through metres first

Worked example

Converting length, mass and capacity

Convert 4.6 km into centimetres, 750 g into kilograms and 1850 cm³ into litres.

Solution:

A centimetre is smaller than a kilometre, so the number of units goes up: multiply
Go through metres: 4.6 km = 4.6 × 1000 = 4600 m
Then 4600 m = 4600 × 100 = 460 000 cm
A kilogram is larger than a gram, so the number of units goes down: divide by 1000
750 g = 750 ÷ 1000 = 0.75 kg
A litre is 1000 cm³, so changing cm³ into litres is changing to the larger unit: divide by 1000
1850 cm³ = 1850 ÷ 1000 = 1.85 litres
Check the sizes: 1 km is 100 000 cm, so 4.6 km must be several hundred thousand centimetres, and 750 g is less than a kilogram

Squared and cubed units

Area units are squared, so the conversion factor is squared too
Volume units are cubed, so the conversion factor is cubed
- 1 cm² = 10² mm² = 100 mm²
- 1 m² = 100² cm² = 10 000 cm²
- 1 km² = 1000² m² = 1 000 000 m²
- 1 cm³ = 10³ mm³ = 1000 mm³
- 1 m³ = 100³ cm³ = 1 000 000 cm³
Picture a square metre as a square measuring 100 cm by 100 cm, which makes the 10 000 obvious rather than something to memorise
The syllabus names conversion between area units and between volume and capacity explicitly, so both are examinable

Worked example

Converting an area

Convert 2.5 m² into cm², and 45 000 cm² into m².

Solution:

One metre is 100 cm, so one square metre is 100 × 100 = 10 000 cm²
Changing to the smaller unit means multiplying
2.5 × 10 000 = 25 000 cm²
Going the other way means dividing by the same factor
45 000 ÷ 10 000 = 4.5 m²
Check: 4.5 × 10 000 = 45 000