BIRKBECK MATHS SUPPORT
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Measurements 1
In this section we will look at
- Examples of everyday measurement
- Some units we use to take measurements
- Symbols for units and converting between units
- Using powers or indices in units
- Converting units with powers or indices
- Combinations of units and how to convert these
Helping you practice
At the end of the sheet there are some questions for you to practice.
Don’t worry if you can’t do these but do try to think about them. This
practice should help you improve. I find I often make mistakes the
first few times I practice, but after a while I understand better.
Online Quizzes and Videos
There are online quizzes and videos covering examples related to
this worksheet. All the answers to questions in this worksheet are
available at the website www.mathsupport.wordpress.com
Good luck and enjoy!
www.mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2011
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1. Measurements
Measurements and units are important in everyday life as well as in health,
engineering and science related jobs. You may need to understand the
difference between ounces and grammes for cooking, kilometres and miles for
driving or pounds and dollars when travelling overseas.
For some jobs understanding measurements and units is a matter of life or
death. If a patient needs 50 milligrammes of a drug and the solution has 250
grammes per litre, how can we calculate how much the patient should drink?
Here we will show how to make measurements and calculations in different
units. Once we’ve learnt how to convert one type of unit, the same rules apply
for all units.
As usual we will start with examples of every day measurements to show how
the maths works then we’ll try some harder examples using the same ideas.
Here are some everyday examples of measurements and units:

The news is on in 10 minutes

The park is about 3 miles away

We need 2 pints of milk

Some chocolate bars contain more than 20 grammes of fat

The chicken must be cooked at 400 degrees Fahrenheit
For each of these examples the measurements are the numbers and the unit
shows what you are measuring. So ‘minutes’ is a unit for measuring time,
‘miles’ is a unit for measuring distance, ‘pints’ is a unit for measures volume,
‘grammes’ is a unit for measuring mass (which is sometime called weight) and
finally ‘degrees Fahrenheit’ is a unit for measuring temperature.
But for each of these measurements we could have used a different unit. For
example we can measure time using the units of hours or weeks. We can
measure distance using the units of metres or miles. We can measure volume
using the units of litres instead of pints, we can measure mass using ounces
and we can measure temperature using degrees Celsius instead of Fahrenheit.
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So the first thing we will look at is how to convert between different units for the
same measurements, here is a quick example
If a caterpillar is 2 inches long, can we convert this to metres?
1 inch is about 2.5 centimetres so we would write this as
1 inch = 2.5 centimetres
So 2 inches is the same as 2 x 2.5 centimetres = 5 centimetres
...and 3 inches = 3 x 2.5 centimetres = 7.5 centimetres
...and 10 inches = 10 x 2.5 centimetres = 25 centimetres
But 'centi' means one hundredth or 0.01, so 1 centimetre = 0.01 metre.
This means 5 centimetres is 5 x 0.01 metre = 0.05 metres
So if we think about our original caterpillar, which is 2 inches long, we can say
2 inches = 2 x 2.5 centimetres = 5 centimetres = 5 x 0.01 metres = 0.05 metres
Don’t worry if you didn’t know that 1 inch is 2.5 centimetres or that 1 centimetre
is 0.01 metre, we will cover all these things and more below. But before we
move on we will quickly introduce some of the important symbols for units
SOME SYMBOLS FOR UNITS
Here are some common units and their symbols
Unit
Symbol
Measurement
Metre
m
length or distance
Second
s
time
Kilogramme
kg
mass (often called weight)
As an example we would write 5 metres as 5m. We would write 30 seconds as
30s and we would write 11 kilogrammes as 11kg.
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SOME SYMBOLS FOR ‘PREFIXES’
A prefix is something that goes in front of a unit. In the word ‘kilometre’ the ‘kilo’
part is the prefix and the ‘metre’ part is the unit. There are lots of prefixes such
as ‘centi’, ‘kilo’, ‘milli’ . Here are the most common
Prefix
Symbol
micro
µ
milli
m
centi
c
deci
d
kilo
k
Mega
M
Giga
G
Amount
The symbol for micro is one of the Greek letters of the alphabet, . This letter is
called mu which rhymes with ‚phew‛ and is pronounced a bit like ‚myou‛.
Combining units and prefixes
It is common to combine prefixes and units, for example, from the two tables
above we can see that a ‘centimetre’ is made up of the prefix ‘centi’ and the unit
‘meter’, so together we would write this as cm which means 0.01 meter.
So 7 millimetres = 7 mm = 7 x 0.001 x metre = 0.007 metres = 0.007m
Combining units together
You will often see the word ‘per’ in combinations of units, for example
kilometres per hour, grammes per litre. The word per is equivalent to dividing.
Here we have written m/s and g/l in three different ways, with a divide sign ( ),
as a fraction and with indices. Each of these ways means the same thing
(metres per second) and the measurements are calculating distance divided by
time (for m/s) and mass divided by volume (for g/l). First though, we will look at
how to convert between different units that measure the same quantity.
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2. Introduction to converting units
We start by converting units of time, length, mass and temperature.
CONVERTING UNITS FOR TIME
We know that there are many ways to measure time; we can go on holiday for
a fortnight which is the same as 2 weeks or fourteen days. So we can write this
mathematically as
1 fortnight = 2 weeks = 14 days
But other units for time include seconds, years, months, hours and we can also
write equations to relate these
1 year = 365 days
1 day = 24 hours = 24 x 60 minutes = 24 x 60 x 60 seconds
So if a plane journey takes 8 hours, this is the same as
8 hours = 8 x 60 minutes = 8 x 60 x 60 seconds = 28,800 seconds
Or if we go on holiday for a fortnight, we can work how many hours this is, as
1 fortnight = 14 days = 14 x 24 hours = 336 hours
CONVERTING UNITS FOR LENGTH
There are lots of units for length, for example
A wall is two meters high
The plane is flying at 37,000 feet
He has a five centimetre scar on his arm
Newcastle is about 200 miles from Dublin
Viruses are about one tenth of a micrometre across
And again we can convert between all these measurements as long as we
know the ‘conversion rate’ between the different units.
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Example 1:
If we want to convert 10 miles to metres and we know that 1 mile is 1.6 km, we
first need to convert miles to kilometres, and then convert kilometres to metres.
So we start with 1 mile = 1.6 km, therefore
But kilo means 1,000, therefore
Or writing this in scientific notation
(Scientific notation and indices are covered in the Numbers and Calculations
worksheets on www.mathsupport.wordpress.com)
Example 2:
To convert 50 kilometres into miles using the fact that 1 mile is 1.6 km then we
have a different problem.
We still have to start with 1 mile = 1.6 km
but this converts from miles → km (as in Example 1) and not from km → miles
(as needed in this example). So we need to find a way to write 1 km in terms of
miles. So we start by swopping sides of the previous equation and we write
If we now divide both sides by 1.6 we get
The left-hand side just gives us 1km (because 1.6 divided by 1.6 is 1), so
Where we used a calculator to re-write
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. We can now calculate
CONVERTING UNITS FOR MASS
It doesn’t matter which units we are converting as the rules are always the
same. But we do need to be a little careful when using the word mass. Most of
the time people use the word weight to mean mass, for example we might say
'the weight of a bag of sugar is about 1 kilogramme'. However some scientists
use weight to mean the force. In fact weight is the force that a mass exerts due
to gravity (so the weight is the force you would feel if someone drops a bag of
sugar on your toe). For this reason we will use the word mass instead of weight.
There are many units for mass (e.g tonnes, stones, grammes or ounces) and if
we know what each unit is in terms of any of the others we can always convert.
Example: If 1 ounce is 28 g, what is 320 grammes in ounces? Well we have
But this only tells us the conversion of ounces → grammes and we need to go
the other way around, ie grammes → ounces, so we need to write
CONVERTING UNITS FOR TEMPERATURE
For cooking and when describing the temperature as part of the weather we
use Celsius
or sometimes Fahrenheit (F). There is another scale for
temperature that engineers or scientists will regularly use called Kelvin (K).
If the temperature is colder than
we use negative numbers, so
means
5 degrees below zero. The coldest temperature possible ‘Absolute Zero’ is
, and this is the start of the Kelvin scale. So
. In general
To convert between Fahrenheit and degrees Celsius we use the formula below,
and we will consider this further in the algebra section.
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3. Units with powers: area and volume
We start by thinking about the meaning of area. One way to do this is to
consider how much carpet we need to cover a floor.
Suppose we have a room that is 5m long and 3m wide and we are fitting
carpet tiles that are each 1m long and 1m wide, we can draw the room as
3m
5m
We can count or calculate the number of tiles needed which is 3 x 5 = 15
This calculation relates to measuring the area of a rectangle and the formula for
area of a rectangle is
But here we are particularly interested in the units. The units we use to
measure length are metres [m] and the units we use to measure the width are
also metres [m]. So to find the units of area, we write
So in this case the area is
.
But we could have measured the length and width in any units measuring
distance (for example feet and inches or centimetres) and the units we use will
affect the results.
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So let’s measure the same area in centimetres. The length 5m is 500cm and the
width of 3m is the same as 300cm, so the diagram of the same room is now
300 cm
500 cm
Using the formula for area of rectangle from above we get
Notice how the unit is now
when previously it was
the area originally calculated as
. Also notice and that
is the same as
. One way to
think about this is to notice that it would take 15 tiles of size 1m by 1m to cover
the floor, but 150,000 tiles of size 1cm by 1cm to cover the same area.
By comparing these two ways of measuring the area (the first one used metres
squared and the second one used centimetres square) we now know that
But rather than drawing out and counting floor tiles every time we want to
convert units, we can use the rules of indices. We start with:
,
,
So 1 square centimetre, which is 1cm x 1cm is written as
Here we have used the rules of indices: if a power is raised to a power we
multiply the powers (in this case -2 x 2 = -4). So we can now see that if
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METERS SQUARED TO FEET SQUARED
Let’s now convert a measurement of area from metres squared (
squared (
) to feet
). We start with the conversion rate of 1 meter = 3.3 feet and draw
our diagram again but with the length and the width in units of feet.
3m = 3 x 3.3 feet
= 9.9 feet
5m = 5 x 3.3 feet
= 16.5 feet
Using the width as 9.9 feet and the length as 16.5 feet, we can now calculate
the area from the formula
to give an answer in units of
But we could arrived at the same answer if we started with
So now we can use our original
to write
So we get exactly the same answer and it doesn’t matter if we draw the
diagram or just convert the unit. However we must be careful and make sure
that if the unit is squared then also square the prefix and the conversion.
Key Point: When converting between units with powers we must always
remember that the powers applying to the units also automatically apply to the
conversion and the prefix.
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Example 1: The International Football Association has set the size of a football
pitch to be 105m long and 68m wide. Calculate the area in meters squared then
using
, convert this into feet squared.
We start by writing out the information we know, then we calculate the area:
Now we want to convert
So we know that
into
, first we calculate
is the same as
the football pitch (which is
in
and we can convert the area of
) into
using
We can double check this answer by calculating the length and width in feet:
,
So we can then calculate the area directly in units of feet squared
Example 2: One tonne of Carbon Dioxide,
temperature occupies a volume of
, at standard pressure and
. Convert this to
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.
VOLUME
Having spent some time on units for areas we will now consider volume.
The volume of an object measures how much space it takes up. Some every
day examples of units for volume include pints and litres. There are lots of
situations when it is important to understand or convert a measurement for
volume. A recipe may need
of water, or a patient may need
of blood or a chemical reaction may need
of acid.
The volume of an object can depend on its shape, and we start with a box with
length 10m, height 5m and width 2m.
height = 5m
width = 2m
length = 10m
The volume of a box is calculated by multiplying the length, width and height.
With any measurement we must always think about units. To measure length,
width and height we use the units meters
. So the units for volume are
And in this case we can calculate the volume as
So we can readily see the unit for volume in this case is
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or ‘metres cubed’
LITRES
Litres are often used to measure volume
and a litre is the volume of a box which
has each side measuring 10cm
10cm
1 Litre
10cm
10cm
CONVERTING BETWEEN LITRES AND METRES CUBED
To convert between litres and
we first notice that 10cm is the same as 0.1m.
This means our volume of 1 litre, which is a box with all sides measuring 10cm,
can also be drawn with all sides 0.1m.
This means:
0.1m
1 Litre
0.1m
But we also notice that
0.1m
which is one decimetre (dm).
This means we can also write
So 1 litre is the same as 1 decimetre cubed.
Examples
So if we know
Example 1: What is
we can calculate the following:
converted to
?
Answer: starting with
we can see that
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Example 2: What is 75ml converted to
?
Answer: the measurement is given in ml (millilitres) so we first need to convert
millilitres to litres.
Now we can use
Example 3: What is
, and carry on from the line above to write
converted to
?
Answer: This question asks us to convert from
to
. All the other
questions so far have asked for
to
and to do these conversions we
used the expression
so we need to find a way to write 1 meter
cubed in litres, so this is where we start
Dividing both sides by
gives us
On the right-hand side we can now cancel the
on the top and the bottom
of the fraction to get 1 and on the left-hand side we can move the denominator
to the numerator to get
, so we can write
This is now our starting point for converting
Example 4: What is
converted to millilitres (
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into litres, and we get
)?
4. Converting units with powers and prefixes
We have seen that 1 litre can be written in many ways,
Notice here that we have changed between
these do not change in the same way as
into
into
into
and into
, and that
. For example
Where as
This is because when the unit is raised to a power (such as meters cubed,
the power applies to the pre-fix as well as the unit. Here is an example
Remember the key point: When converting units with powers, the powers
applying to the units also apply to the prefixes and the conversion factor.
Examples:
For these examples you should only need the prefix table on page 4, but if you
want to revise indices and powers, see the Numbers page of the website
www.mathsupport.wordpress.com
Example 1: Convert
Answer: we start by using
so we can write
If we now want to write this in scientific notation, we write
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as
)
Example 2: Convert
Answer: here we need to use
but we are converting from meters to
millimetres, so we need an expression for meters in terms of millimetres
If we now divide both sides by
we get
so we can write an expression for meters in terms of millimetres
Now with this as our starting point we can convert
in the following way
Which is the answer and it is already in scientific notation.
Example 3: Convert
Starting with
into
we can write
Where we have shown all the steps in one line and given our answer in
scientific notation,
Example 4: Convert
into
Starting with
Example 5: Convert
we can write
into
Starting with
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5. Introducing combinations of Units
So far we have considered units such as meters (for distance), seconds (for
time) and grammes (for mass). We have also considered the meaning of units
with prefixes, such centimetres (
) or milligrammes (
units with powers (volume is measure in
) and explored
and area is measured in
).
We have also considered other ways to take measurements, such as using
litres to measure volume (
) or Fahrenheit or Kelvin to measure
temperature. There are lots more examples with converting units in the later
worksheets and we shall try exercises at the end of this worksheets. But before
we finish we will look at combinations of units.
SPEED
A good example of a using a combination of units occurs when measuring
speed. If you drive a car you measure your speed in miles per hour. This unit
‘miles per hour’ measures the distance [miles] per time [hours], so
So the word ‘per’ means ‘divided by’ and more generally we can write
But we know that distance can also be measured in meters, and time can also
be measured in seconds, so from the above formula we can also measure
speed in
or
or
. We will explore these ideas in the
next worksheet.
DENSITY
Another common measurement with a combination of units is density. The unit
for density is kilogramme per meter cubed or
and the formula is
We will look at how to convert measurements when we have a combination of
units (such as speed or density) in the next section.
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Now your turn
Generally the more maths you practice the easier it gets. If you make mistakes
don’t worry. I generally find that if I make lots of mistakes I understand the
subject better when I have finished. If you want to see videos explaining these
ideas and showing the answers visit www.mathsupport.wordpress.com
A) State whether the following are measuring volume, mass, time,
temperature or distance
1) 40 milligrams of vitamin E
5) 196 Kelvin
2) 75g of flour
6) A journey of 14 miles
3) A prescription for 25ml
7) A journey of 37 minutes
4) 147 milliseconds
8) A rubbish skip sized 8 cubic yards
B) Convert the units with pre-fixes listed below, you may want to use the
table in section 1
1) 50 g into kg
5) 120 ml into litres
2) 5 hours into minutes
6) 0.72 kg into g
3) 0.4 seconds into milliseconds
7) 120cm into m
4) 5 hours into minutes
8) 2.53m into mm
C) Convert the units listed below, using the information: 1 kilometre = 0.62
miles, 1 ounce = 28 gramme, 1 litre =
, 1 inch = 2.5 cm
1) 14 kilometers → miles
5) 2.1 miles → kilometres
2) 15 grammes → ounces
6) 7 ounces → grammes
3) 2 litres → metres cubed
7) 4 metres cubed → litres
4) 3 inches → metres
8) 21 cm → inches
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D) Converting between units with different prefixes
1)
5)
2)
6)
3)
7)
4)
8)
E) Converting between measurements using units with powers
1)
5)
2)
6)
3)
7)
4)
8)
F) Converting between measurements using different units, with prefixes
and powers in the units, using information: 1 inch = 2.5 cm, 1 metre = 3.3 feet
1)
5)
2)
6)
3)
7)
4)
8)
G) General word problems involving units.
1) The area of Singapore is
. If 1 mile is 1.6 km, what is the area of
Singapore in square miles?
1) A typical jet engine takes in 1.2 tons of air per second during takeoff. If a ton
is 40 cubic feet and 1 metre is 3.3 feet. What is the volume of air per second in
metres cubed per second?
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