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S8 Measures
KS4 Mathematics
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Contents
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S8.1 Converting units
S8 Measures
S8.3 Calculations involving bounds
S8.4 Compound measures
S8.5 earings
S8.2 !ccurac" in measurement
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Converting units
It is important to be able to convert between a variety o
units !uickly and accurately"
#or e$ample% i we are usin& a ormula to ind t'e volume
o an ob(ect we must make sure t'at all t'e len&t's are
written usin& t'e same units beore usin& t'e ormula"
)* #hen using a formula
2* #hen comparing measurements
#or e$ample% suppose one mot'er &ives t'e wei&'t o 'er
baby in pounds and anot'er mot'er &ives t'e wei&'t o 'er
baby in kilo&rams"
+e usually use metric units or calculations"
,ow can we compare t'e babies- wei&'ts.
/o compare t'e wei&'ts we convert t'em to t'e same unit"
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Metric units
/'e metric system o measurement is based on powers o
ten and uses t'e ollowin& prei$es
/'ese prei$es are t'en ollowed by a base unit"
/'e base unit or len&t' is metre"metre.
/'e base unit or mass is &ram"gram.
/'e base unit or capacity is litre"litre.
Kilo$
Centi$
Milli$Micro$
meanin& one t'ousand
meanin& one 'undredt'
meanin& one t'ousandt'meanin& one milliont'
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Metric units of length% mass and capacit"
1ou s'ould know t'e ollowin& metric conversions or
len&t'% mass and capacity li!uid volume*
&ength
&ength
1 'm ( 1))) m
1 m ( 1)) cm
1 m ( 1))) mm
1 cm ( 1) mm
Mass
Mass
1 tonne ( 1))) 'g
1 'g ( 1))) g
1 g ( 1))) mg
Capacit" and
*olume
Capacit" and
*olume
1 litre ( 1))) ml
1 cl ( 1) ml
1 m3 ( 1))) litres
1 cm3 ( 1 ml
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Metric units of area and volume
/'e metric units or area are mm2% cm2% m2 and km2"
) m2 3 )00 cm 4 )00 cm
) m2 3 )0%000 cm2
) m2 3 )0%000 cm2
) cm2 3 )0 mm 4 )0 mm
) cm2 3 )00 mm2
) cm2 3 )00 mm2
/'e metric units or volume are mm% cm% m and km"
) m 3 )00 cm 4 )00 cm 4 )00 cm
) m 3 )%000%000 cm
) m 3 )%000%000 cm
) cm 3 )0 mm 4 )0 mm 4 )0 mm
) cm 3 )000 mm
) cm 3 )000 mm
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Converting metric units
/o convert rom a lar&er metric unit to a smaller
one we need to ------- by )0% )00% or )000"
/o convert rom a lar&er metric unit to a smaller
one we need to ------- by )0% )00% or )000"multipl"
#or e$ample%
0"6 k& 3 0"6 4 )000 &
7onvert 0"6 k& to &rams
3 60 &
/o convert rom a smaller metric unit to a lar&er
one we need to ------- by )0% )00% or )000"
/o convert rom a smaller metric unit to a lar&er
one we need to ------- by )0% )00% or )000"divide
#or e$ample%
"9 cm 3 "9 : )00 m
7onvert "9 cm to metres
3 0"09 m
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Converting metric units
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/rdering units
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0mperial units
Imperial units are still re!uently used and you s'ould be
aware o t'e ollowin& imperial conversions
) oot 3 ;;; inc'es)2
) yard 3 ;;; eet
) pound 3 ;;; ounces)9
) stone 3 ;;; pounds)6
) &allon 3 ;;; pints8
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Converting imperial units
+e can convert between metric and imperial units usin& t'e
ollowin& appro$imate conversions
5 miles is about ;;; kilometres8
) oot is about ;;; centimetres0
) inc' is about ;;; centimetres2"5
) kilo&ram is about ;;; pounds2"2
) &allon is about ;;; litres6"5
) litre is about ;;;; pints)"5
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Converting imperial units
) kilo&ram is about 2"2 pounds"
About 'ow many kilo&rams are t'ere in ) stone.
) stone 3 )6 pounds
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Metric and imperial conversions
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Spider diagram
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Contents
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S8.2 !ccurac" in measurement
S8.1 Converting units
S8.3 Calculations involving bounds
S8.4 Compound measures
S8.5 earings
S8 Measures
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Continuous measurements
/'e continuous nature o measurements means t'at t'ey
can never be e$act" /'ere is always an element oroundin& involved"
I we were measurin& t'e len&t' o a pencil% or e$ample% we
would measure to t'e nearest cm or mm"
/'ese t'ree pencils all measure )6 cm to t'e nearest cm
13 14 15 13 14 15 13 14 15
/'e len&t' o a pencil &iven as )6 cm to t'e nearest cm can
be any len&t' between )"5 cm and )6"5 cm"
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!ccurac" in measurement
>emember% any measurement &iven to t'e nearest w'ole
unit could be up to 'al a unit lon&er or s'orter"
>emember% any measurement &iven to t'e nearest w'ole
unit could be up to 'al a unit lon&er or s'orter"
Suppose we are told t'at a pencil measures )6"2 cm"
7an we assume t'at t'is measurement is e$act.
?o% alt'ou&' t'is measurement 'as been &iven to a 'i&'er
de&ree o accuracy it is not e$act"
+'at is t'e s'ortest and lon&est len&t' it could be.
/'e len&t' l o a pencil &iven as )6"2 cm% to t'e nearest 0")
cm% could be anyw'ere in t'e ran&e
)6")5 cm @ l )6"25 cm)6")5 cm @ l )6"25 cm
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pper and loer bounds
+'en we &ive a ran&e or a measurement as in
)6")5 cm @ len&t' )6"25 cm
t'is value is calledt'e lower bound t'is value is calledt'e lower bound
and t'is value iscalled t'e upper bound" and t'is value iscalled t'e upper bound"
/'e len&t' could be e!ual to )6")5 cm so we use a &reater
t'an or e!ual to symbol"
I t'e len&t' was e!ual to )6"25 cm 'owever% it would 'ave
been rounded up to )6" cm" /'e len&t' is t'ereore Cstrictly
less thanD )6"25 cm and so we use t'e symbol"
/'is is an
ineualit"
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pper and loer bounds
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Contents
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S8.3 Calculations involving bounds
S8.1 Converting units
S8.2 !ccurac" in measurement
S8.4 Compound measures
S8.5 earings
S8 Measures
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!dding measures
/'e ollowin& trian&le 'as sides o len&t' cm% 6 cm and 5 cm"
cm5 cm
6 cm
+'at is t'e ran&e o possible
len&t's or t'e perimeter.
/'e least t'e len&t's could be is 2"5 cm% "5 cm and 6"5 cm"
/'e smallest possible perimeter 3 2"5 E "5 E 6"5 3 )0"5 cm
/'e most t'e len&t's could be is "5 cm% 6"5 cm and 5"5 cm"
/'e lar&est possible perimeter 3 "5 E 6"5 E 5"5 3 )"5 cm
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!dding measures
/'e ollowin& trian&le 'as sides o len&t' cm% 6 cm and 5 cm"
cm5 cm
6 cm
+'at is t'e ran&e o possible
len&t's or t'e perimeter.
/'e ran&e o possible values or t'e perimeter is
?otice t'at t'e more len&t's t'at are added to&et'er t'e
&reater t'e error on eit'er side"
)0"5 cm @ perimeter )"5 cm)0"5 cm @ perimeter )"5 cm
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!dding measures
+'en a calculation involves adding two or more
measurements to&et'er
/'e lower bound is ound by
addin& t'e lower bounds to&et'er addin& t'e lower bounds to&et'er
/'e upper bound is ound by
addin& t'e upper bounds to&et'er addin& t'e upper bounds to&et'er
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Calculations involving bounds
A piece o wood measurin& )0 cm 'as
a piece o len&t' 50 cm cut o o it"
/'ese are &iven to t'e nearest )0 cm"
+'at is t'e ran&e o possible
siFes or t'e remainin& piece.
/'e ori&inal piece o wood could be between )95 cm and
)5 cm"
/'e piece o wood t'at is cut o could be between 65 cmand 55 cm"
/'e smallest possible siFe o t'e remainin& piece is
)95 G 55 3 ))0 cm
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Calculations involving bounds
A piece o wood measurin& )0 cm 'as
a piece o len&t' 50 cm cut o o it"
/'ese are &iven to t'e nearest )0 cm"
+'at is t'e ran&e o possible
siFes or t'e remainin& piece.
/'e ori&inal piece o wood could be between )95 cm and
)5 cm"
/'e piece o wood t'at is cut o could be between 65 cmand 55 cm"
/'e lar&est possible siFe o t'e remainin& piece is
)5 G 65 3 )0 cm
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Calculations involving bounds
A piece o wood measurin& )0 cm 'as
a piece o len&t' 50 cm cut o o it"
/'ese are &iven to t'e nearest )0 cm"
+'at is t'e ran&e o possible
siFes or t'e remainin& piece.
/'e ori&inal piece o wood could be between )95 cm and
)5 cm"
/'e piece o wood t'at is cut o could be between 65 cmand 55 cm"
/'e ran&e o possible siFes or t'e remainin& piece is
))0 cm @ len&t' )0 cm))0 cm @ len&t' )0 cm
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Calculations involving bounds
+'at is t'e ran&e o possible values or t'e an&le a.
/'e smallest a could be is )80H G 5"5H E 8"5H* 3
/'e ran&e o possible values or a is t'ereore%
99H @ a 98H 99H @ a 98H
/'e an&les in t'e ollowin& dia&ram are rounded to t'e
nearest de&ree
99H
5H 8Ha
/'e lar&est a could be is )80H G 6"5H E "5H* 3 98H
/'is an&le could
be between 6"5H
and 5"5H"
/'is an&le could
be between "5H
and 8"5H"
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Subtracting measures
+'en a calculation involves subtracting two measurements
/'e lower bound is ound by
subtractin& t'e upper bound rom t'e lower boundsubtractin& t'e upper bound rom t'e lower bound
/'e upper bound is ound by
subtractin& t'e lower bound rom t'e upper boundsubtractin& t'e lower bound rom t'e upper bound
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Calculations involving bounds
/'e dimensions o a small tile are
&iven as )5 mm by )8 mm"
)5 mm
)8 mm
+'at is t'e smallest possible
area t'e tile could 'ave.
/'e smallest values or t'e len&t' and t'e widt' o t'e tile
are )6"5 mm by )"5 mm"
/o calculate t'e smallest possible area we multiply t'ese
values to&et'er"
Smallest possible area 3 )6"5 4 )"5 3 25"5 mm2
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Calculations involving bounds
/'e dimensions o a small tile are
&iven as )5 mm by )8 mm"
)5 mm
)8 mm
+'at is t'e lar&est possible
area t'e tile could 'ave.
/'e lar&est values or t'e len&t' and t'e widt' o t'e tile are
)5"5 mm by )8"5 mm"
/o calculate t'e lar&est possible area we multiply t'ese
values to&et'er"
Lar&est possible area 3 )5"5 4 )8"5 3 289"5 mm2
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Calculations involving bounds
/'e dimensions o a small tile are
&iven as )5 mm by )8 mm"
)5 mm
)8 mm
+'at is t'e ran&e o possible
values or t'e area.
/'e smallest value or t'e area is 25"5 mm2 and t'e
lar&est value or t'e area is 289"5 mm2
/'e ran&e o possible values or t'e area is t'ereore%
25"5 mm2 @ area 289"5 mm225"5 mm2 @ area 289"5 mm2
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Multipl"ing measures
+'en a calculation involves multipl"ing two or more
measurements to&et'er
/'e lower bound is ound by
multiplyin& t'e lower bounds to&et'er multiplyin& t'e lower bounds to&et'er
/'e upper bound is ound by
multiplyin& t'e upper bounds to&et'er multiplyin& t'e upper bounds to&et'er
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Calculations involving bounds
A boy runs 200 metres in 2"8 seconds"
+'at is 'is &reatest possible
avera&e speed to 2 decimal places.
/'e distance is &iven to t'e nearest metre and t'e time is
&iven to t'e nearest tent' o a second"
/'e distance could be between )5"5 m and 200"5 m"
/'e time taken could be between 2"5 s and 2"85 s"&reatest possible
avera&e speed3
&reatest distance
s'ortest time3
200"5 m
2"5 s3 "2 mJs
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ividing measures
+'en a calculation involves dividing two measurements
/'e lower bound is ound by
dividin& t'e lower bound by t'e upper bounddividin& t'e lower bound by t'e upper bound
/'e upper bound is ound by
dividin& t'e upper bound by t'e lower bounddividin& t'e upper bound by t'e lower bound
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Contents
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S8.4 Compound measures
S8.1 Converting units
S8.2 !ccurac" in measurement
S8.5 earings
S8.3 Calculations involving bounds
S8 Measures
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Compound measures
+'ic' is 'eavier a kilo&ram o cotton wool
or a kilo&ram o rocks.
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Compound measures
K course% a kilo&ram o rocks wei&'s t'e same as a
kilo&ram o cotton wool G t'ey bot' wei&' one kilo&ram
I you t'ou&'t t'at t'e rocks were 'eavier% you were probably
t'inkin& o t'eir density rat'er t'an t'eir wei&'t"
ensity is an e$ample o a compound measure"
It is a measure o t'e mass o an ob(ect per unit volume"
ensity is usually measured in &Jcm or k&Jm" It can also
be measured in k&Jl "
A density o )"2 &Jcm means t'at every ) cm o t'e material
'as a mass o )"2 &rams"
C d
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Compound measures
/'is is a more sensible !uestion" >ock is denser t'an cotton
wool and so ) cm o rock is 'eavier t'an ) cm o cotton wool"
+e can ind t'e density o a &iven material usin& t'e
ollowin& ormula
ensity 3mass
volume
+'ic' is 'eavier ) cm o cotton wool or ) cm o rock.
#or e$ample% i a block o metal 'as a mass o 90 & and avolume o 80 cm
ensity 390
803 "5 &Jcm
C d
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Compound measures
+e use compound measures w'en we are comparin& 'ow
one measurement c'an&es wit' anot'er"
+'en one measurement c'an&es in direct proportion wit'
anot'er it is said to c'an&e at a constant rate"
#or e$ample% suppose a man is runnin& around a race track"
/'e total distance 'e 'as run c'an&es wit' time"
/'e rate at w'ic' 'e runs is called 'is speed"
Speed is usually measured in kmJ'% mJs or mp'"
Speed 3 distance travelledtime taken
! d
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!verage speed
In many situations t'e speed is not constant"
#or e$ample% t'e man runnin& around t'e track will probably
speed up or slow down as 'e runs"
+e can still calculate 'is avera&e speed usin& t'e ollowin&
ormula
Avera&e speed 3total distance travelled
total time taken
#or e$ample% i t'e man runs )590 metres in 00 seconds
Avera&e speed 3)590
003 5"2 mJs
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Common compound measures
7ommonly used compound measures include
#uel consumptionMeasured in kmJl or
mp&"
NressureMeasured in ?Jm2 or
?Jcm2"
SpeedMeasured in mJs% kmJ'
or mp'"
ensityMeasured in &Jcm%
k&Jm or k&Jl "
mass
volume
distance
time
orce
surace area
distance
volume
i d
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airs 6 compound measures
C l l ti d iti
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Calculating densities
C ti d it
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Converting compound units
+'at is 65 mp' in mJs.
+'en we convert compound units we usually 'ave to do it in
several steps% especially i bot' units are bein& c'an&ed"
65 mp' stands or 65 miles per 'our"
+e 'ave to c'an&e t'e 65 miles into metres and ) 'our into
seconds" 5 miles 3 8 km
So ) mile 3 )"9 km65 miles = 65 4 )"9 km
3 2 km3 2%000 m
) 'our 3 90 minutes
3 900 seconds
C ti d it
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Converting compound units
+'at is 65 mp' in mJs.
/ravellin& 65 miles in ) 'our is e!uivalent to travellin& 2 000 m
in 900 seconds"
/'ereore% 65 mp' =2%000
900mJs
3 20 mJs
,ow could we use t'is answer to convert any
speed &iven in mp' to a speed in mJs.
65 mp' = 20 mJs
ividin& by 5 we 'ave% mp' = 6 mJs
+e can divide t'e speed in mp' by and multiply it by 6"+e can divide t'e speed in mp' by and multiply it by 6"
Contents
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Contents
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S8.5 earings
S8.1 Converting units
S8.2 !ccurac" in measurement
S8.4 Compound measures
S8.3 Calculations involving bounds
S8 Measures
earings
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earings
earings are a measure o direction taken rom ?ort'"
I you were travellin& ?ort' you would be travellin& on a
bearin& o 000H"
I you were travellin& rom t'e point N in t'e direction
s'own by t'e arrow t'en you would be travellin& on abearin& o 000H"
I you were travellin& rom t'e point N in t'e direction
s'own by t'e arrow t'en you would be travellin& on abearin& o ),57.
?
5H
N
Bearin&s are always measured
clockwise rom ?ort' and arewritten as t'ree i&ures"
Compass points
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Compass points
N
E W
S
NE
SE SW
NW
000H
065H
00H
)5H
)80H
225H
20H
)5H
Measuring bearings
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Measuring bearings
ac' bearings
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ac' bearings
/'e bearin& rom point A to point B is )05O"
+'at is t'e bearin& rom point B to point A.
A
?
)05O
B
?
.
/'e an&le rom B to A is
)05O E )80O 3 285
/'is is called a bac'
bearing")05O
)80H
0nvestigating bac' bearings
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0nvestigating bac' bearings