fractions, decimals and percentages - pearson education · • convert between fractions, decimals...

$: Fractions, decimals and percentages - Pearson Education · • Convert between fractions, decimals and percentages • Compare fractions, decimals and percentages • Write a fraction$
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4.1 Equivalent proportionsYou will learn to:• Convertbetweenfractions,decimalsandpercentages• Comparefractions,decimalsandpercentages• Writeafractionasadecimal.

Master extend P99

test P103

Check P93

strengthen P95

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4 Fractions, decimals and percentages

Exercise 4.11 Writeasapercentage.

a 72outof 100 b 25outof 50 c 0.73 d 1.29

2 Workouta 34÷10 b 286÷1000 c 9021÷100

3 Writeeachdecimalasafraction.a 0.75 b 0.3 c 0.9

4 Write<or>betweeneachpairof decimals.a 0.15u0.172 b 0.62u0.603 c 4.049u4.5

5 Sorttheseintofivesetsof equivalentproportions.

0.75

5%

17100

0.05

80%

710

0.17

17%

34

0.7

70%

45

0.8

75%

120

6 Writeeachfractionasadecimalandasapercentage.

a 1__20=0.u=u% b 3__20=0.u=u%

c 3_8=0.u=u% d 7__20=0.u=u%

e 7_8=0.u=u% f 11__20= 0.u=u%

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Key pointYou can write a fraction as a decimal by dividing the numerator by the denominator. 2 _ 4 = 2 ÷ 4 = 0.5

Q5 hint

Rewrite fractions so they have denominator 100 to work out the percentage.

Q6 hint

Use the S D key on your calculator to change the fraction into a decimal.

topic links: Measures

Fluency What is the decimal equivalent of• 1 _ 2

• 3 _ 4

• 1 __ 10

• 7 __ 10

Why learn this?Information about proportions can be given as fractions, decimals or percentages. You need to be able to convert between different types.

explore Can you write the number π as a fraction?

80Unit 4 Fractions, decimals and percentages

4 Fractions, decimals and percentages

7 Writeeachfractionasadecimalandasapercentage.

a 32__40=0.u=u% b 71__80=0.u=u%

c 90___120=0.u=u% d 105___150=0.u=u%

e 11__16=0.u=u% f 18___125=0.u=u%

8 Problem-solving Whichshapehasthelongerperimeter?

0.1 m

m

0.1 m

0.3 m

0.4 m

A B 320

m720

m120

m820

9 Usethedecimalequivalentof eachfractiontowritethesetsinorder,smallesttolargest.

a 13__20 3_5 5_8 b 3_8 7__20 16__40

10 Putthesevaluesinorder,smallesttolargest.13__20 0.62 64.5% 9outof 16

Worked exampleWrite0.245asafraction.

0.245 = 245 ______ 1000

= 49 ____ 200

11 Writeeachdecimalasafraction.Simplifywherepossible.

a 0.85 = 85 ____ 100

= b 0.375 = u _____

1000 =

c 0.84 d 0.125e 0.23 f 0.875g 0.19 h 0.444

12 Writeeachpercentageasafraction.Simplifywherepossible.

a 35%= 0.35 = u ____

100 =

u ___

20 b 6%=

u ____

100 =

u __

u

c 88% d 5%

e 12.5%= 0.125 = 125 _____ 1000

= f 37.5%

g 45.8% h 1.2%

13 Explore Canyouwriteπasafraction?Choosesomesensiblenumberstohelpyouexplorethissituation.Thenusewhatyou’velearnedinthislessontohelpyouanswerthequestion.

14 Reflect Thislessonusesalotof mathematicalterms,suchas•terminating•equivalent•percentage•commonfactor.Writedownwhateachof thesetermsmeans,inyourownwords.Whichtermsarenew,andwhichoneshaveyoumetbefore?

Key pointA terminating decimal ends after a definite number of digits, for example 0.39 and 1.042.You can write any terminating decimal as a fraction.

There are 3 decimal places so 245 has been divided by 1000

Divide numerator and denominator by 5 to simplify.49 and 200 don’t have any common factors so it cannot be simplified further.

Q11 hint

Use the number of decimal places to decide whether the denominator should be 100 or 1000.

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Pi 3, Section 4.1

81

Exercise 4.21 Workout

a 726÷6 b 7125÷5 c 345÷4 d 842÷8

2 a i Jarredthinksof anumber.Half of itis240.Whatis1_4of hisnumber? ii WhatnumberdidJarredthinkof?

b i Sophiethinksof anumber.1_6of itis12.Whatis1_3of hernumber?

ii WhatnumberdidSophiethinkof?c i Ameethinksof anumber.1__10of itis250.Whatis

1_5of hernumber? ii WhatnumberdidAmeethinkof?

3 Reasoning a Useyourcalculatortomatcheachfractiontoitsequivalentdecimal.

35

58

59

23

712

711

0.625 0.5 0.6 0.6 0.63 0.583

b Usethedecimalstoorderthefractionsinparta,smallesttolargest.

Worked exampleWrite1_9asadecimal.

1.1000 __ 9)

0

1.10101010 __

9) 0.1111…

1 __ 9 = 0. 1 ˙

4 Writeeachfractionasadecimal.

a 1_8 b 1__12 c 5__12 d 7_8

e 1_6 f 5_6 g 2_9 h 8_9

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Q2b hint

16

16

13

12

Key pointA recurring decimal contains a digit, or sequence of digits, which repeats itself forever. A dot over the digit shows it recurs. For example, 0.111 11…. = 0. 1 ˙

9 doesn’t go into 1 so write a 0 in the units column.

There are now 10 tenths.

9 goes into 10 once with remainder 1.There are now 10 hundredths. Continue like this and the decimal recurs.

topic links: Division, Pie charts

4.2 Recurring decimalsYou will learn to:• Writerecurringdecimalsasfractions.

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FluencyRound each number to 2 decimal places.• 1.454• 6.087• 3.3642

• 10.4985

explore How can you tell if a fraction will be a terminating or a non-terminating decimal?

Check P93

extend P99

strengthen P95

test P103

Why learn this? Lots of calculations in real life have an answer that is a recurring decimal.

Master


5 Problem-solving / Reasoning Niraworksout12÷18andgets

ananswerof 0.6666667 onhercalculator.

Whatistheequivalentfraction?Discussion Whathasthecalculatordone?

6 Useawrittenmethodtoworkouteachdivision.Writeyouranswersasrecurringdecimalsusingdotnotation.a 823÷3 b 375÷9 c 37564÷3d 6385÷9 e 97÷12 f 1756÷12

Investigation Reasoning / Problem-solvingCaroline says, ‘Some fractions can be written as terminating decimals but some are recurring decimals’.1 Write each fraction as a decimal.

1 _ 2 1 _ 3 1 _ 4 1 _ 5 1 _ 6 1 _ 7 1 _ 8 1 _ 9 1 __ 10 1 __ 11 1 __ 12 2 Sort them into terminating decimals and recurring decimals. 3 Jack says, ‘If the denominator of a fraction is even, it will be a recurring decimal’.Find an example to show that Jack is wrong.4 Which denominators give terminating decimals?Discussion Caroline says, ‘I think it’s to do with the fact that 2 × 5 = 10.’ What do you think she means?5 Investigate what happens if the numerator is not 1.

7 a Whichbaghasthegreaterproportionof redcounters?b Whatistheproportionof bluecountersineachbag?

A B

8 Thepiechartshowsthefirstlanguageof peopleworkinginasummerschool.Writetheproportionof eachlanguageasafractionandasapercentage.

9 Explore Howcanyoutellif afractionwillbeaterminatingoranon‑terminatingdecimal?Isiteasiertoexplorethisquestionnowyouhavecompletedthelesson?Whatfurtherinformationdoyouneedtobeabletoanswerthis?

10 Reflect Ashorthandwayof writingarecurringdecimalistouseadot,ordots,overthenumbersthatrepeat.Forexample,0.2222…iswrittenas0.2˙ Writedownfiveothershortformsthatyouuseinmaths.Doyouthinktheseshortformsareusefulornot?

150

3060

120

Chinese

International School Languages

English

French

Russian

Q8 hint

How many degrees represent the whole pie chart?

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Pi 3, Section 4.2

83

Master

4.3 Adding and subtracting fractionsYou will learn to:• Addandsubtractfractions• Addandsubtractmixednumbers.

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Fluency What is• 1 _ 2 + 1 _ 4

• 1 _ 2 – 1 _ 4

• 3 _ 5 – 3 __ 10

• 3 _ 8 + 1 _ 2

explore How many years ago did people start writing fractions?

Check P93

extend P99

strengthen P95

Test P103

Why learn this?Statisticians add and subtract fractions to work out the probably of different events happening (or not happening).

Exercise 4.31 Writeeachimproperfractionasamixednumber.

a 14__5 b 20__8 c 12__7 d 23__3

2 Writeeachmixednumberasanimproperfraction.a 21_4 b 12_3 c 53_8 c 43__10

3 Workouta 5__12+

1_3 b 5_6–1_3 c 3_4–

3_8 d 1_3+2_5

4 Workouteachcalculation.Giveyouranswerasamixednumberwherenecessary.

a 5_8+3_8+

1_8 b 7__12+1__12+

11__12 c 5_9–2_9+

7_9

d 1_2–1_3+

1_4 e 4_5+3__10–

3_4 f 2_3–4_9+

1_6

5 Workout

a 1_2+1_3+

1_4 b 1_3+1_4+

1_5 c 1_4+1_5+

1_6

6 Problem-solving Workoutthemissingnumber.a 4_5+

3_4+u=2 b 5_6+3_5–u=1

7 Problem-solving Inaclothesshop,1_5of clothesaresuits,2_3are

trousersandtherestaretops.Whatfractionof theclothesaretops?

Worked examplea Workout35_6+1

3_4

3 5 _ 6 + 1 3 _ 4 = (3 + 1) + ( 5 _ 6 + 3 _ 4 )

= 4 + ( 10 __ 12 + 9 __ 12 )

= 4 + 19 __ 12

= 4 + 1 7 __ 12

= 5 7 __ 12

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Q4a hint

5 1

18

18

18

18

18

18

18

18

18

88

58

38

18

Add the whole number parts and add the fraction parts separately.

Convert the fractions to equivalent fractions with a common denominator.

Change the improper fraction to a mixed number so you can add the whole number parts.


b Workout21_2–15_6

2 1 _ 2 – 1 5 _ 6 = (2 – 1) + ( 1 _ 2 – 5 _ 6 ) = 1 + ( 3 _ 6 – 5 _ 6 )= 1 + (– 2 _ 6 )= 1 – 1 _ 3 = 2 _ 3

8 Workout

a 11_5+23__10 b 21_4+3

2_5 c 21_3+22_5

d 34_5+11_3 e 42_3+2

5_6 f 27__10+34_5

g 111__12+15_6 h 33_4+1

4_5 i 25_9+35_6

9 Workout

a 27__10–12_5 b 23_4–1

3_8 c 45_6–32_3

d 34_5–11_3 e 42_3–2

1_6 f 37__10–33_5

g 411__12–25_6 h 33_4–1

3_5 i 35_6–25_9

10 Yazdiuses21_4litresof whitepaintand23_5litresof bluepaint.Howmany

litresof paintdidheuseintotal?Giveyouranswerasamixednumber.

11 Peterwalked45_6km,Brendawalked34_5km.HowmuchfurtherdidPeter

walk?

12 Afarmerneeds32_5metresof nettingforhischickens.Healreadyhas111__20metres.Howmuchmoredoesheneed?

13 Aboxof applesweighs35_7kgandaboxof pearsweighs23_5kg.How

muchdotheyweighaltogether?

Investigation Reasoning1 Write down six fractions that are between

a 0 and 1 b 0 and 1 _ 2 c 1 _ 2 and 1

0 112

2 Write down two fractions that are between

a 1 _ 4 and 3 _ 4 b 2 _ 8 and 3 _ 8 c 7 __ 10 and 8 __ 10 d 3 _ 5 and 4 _ 5

Discussion In how many different ways can you answer the questions in this investigation?

14 Explore Howmanyyearsagodidpeoplestartwritingfractions?Whatinformationdoyouneedtostartansweringthisquestion?

15 Reflect Chooseoneof thepartsof Q9thatyoufeltconfidentinanswering.Howwouldyouexplainthemethodyouusedtoaclassmatewhohadmissedthislesson?

Subtract the whole number parts and the fraction parts separately.

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Pi 3, Section 4.3

85


a 3×1_4 b 2×3_7 c 2_5×2

d 4×2_9 e 5_6×4 f 1_3×6

2 Writeeachimproperfractionasamixednumberinitssimplestform.

a 32__10 b 22__4

c 35__5 d 34__6

3 Whatistheinverse operationof

a multiplyingby10 b dividingby8?

4 Simonesays,‘I’mthinkingof anumber.Imultiplyitby8andthendividetheanswerby8,andget5.’Whatnumberwasshethinkingof?

5 Reasoning Workout

a 8×1÷4 b 8×1_4 c 8÷4×1d Explainwhytheansweristhesameinpartsa,bandc.

6 Workout

a 6×1_3 b 5×4_5

c 2_3×3 d 5_7×7

7 Workout

a 2_5×250 b 2_3×360 c 2_3of 360

8 Real / Problem-solving Acarhas45litresof fuelinthetank.Thedriveruses3_5of thefuel.Howmanylitresof fuelareleft?

9 Reasoning Whichof theseproductswillhaveananswerlessthan1?

a 5×2_3 b 1_9×7

c 2×4__15 d 3_4×6

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Q3 Literacy hint

An inverse operation is the opposite operation.

Q6b hint

Work out one-fifth of 5, then multiply it by 4.

Q7a hint

Work out one-fifth of 250, then multiply by 2.

Master

4.4 Multiplying fractionsYou will learn to:• Usestrategiesformultiplyingfractions.

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FluencyWhat is• 3 _ 7 of 21

• 2 _ 5 of 15

• 2 _ 9 of 36

• 5 _ 6 of 72

explore How many times can you cut a piece of paper in half?

Check P93

extend P99

strengthen P95

test P103

Why learn thisYou multiply fractions when working out the distance you can travel with half a tank of fuel.


Worked example Workout1_4×

2_3

1 _ 4 × 2 _ 3 = 1 × 2 ____ 4 × 3

= 2 __ 12

= 1 _ 6

10 Workouteachcalculation.Simplifyyouranswerwhereneeded.

a 2_3×3_4=

2 × 3 ____ 3 × 4

= u

__ u

= u

__ u

b 1_4×4_5 c 2_3×

2_5

d 3_4×3_4 e 3_4×

6__11 f 4_9×3_7 g 3_5×

7__12

11 Workout

a 1_2×1_4 b 1_2×

1_3 c 1_2×1_5 d 1_2×

1_2

Discussion Whathappenstothedenominatorwhenyoumultiplyafractionby1_2?

12 Workout

a (1_4)2=1 _ 4 × 1 _ 4 = b (1_3)

2 c (1_5)2

d (1_6)2 e (2_3)

2 f (3_8)2

Worked example Workout3_8×

2_9

3 _ 8 × 2 _ 9 = 3 × 2 _____ 8 × 9

= 2 × 3 _____ 8 × 9

= 2 _ 8 × 3 _ 9

= 1 _ 4 × 1 _ 3

= 1 __ 12

Discussion Howcouldyouworkoutthemultiplicationusingfewersteps?

13 Workout

a 5_8×3_5 b 3_4×

2_7 c 3_4×8__15

d 4_9×3_8 e 9__15×

5_6 c 5_6×3

__20

14 Hollydrank2_3of a1_2litrebottleof juice.Howmuchdidshedrink?

15 Explore Howmanytimescanyoucutapieceof paperinhalf?Choosesomesensiblenumberstohelpyouexplorethissituation.Thenusewhatyou’velearnedinthislessontohelpyouanswerthequestion.

16 Reflect LookagainatQ8.Writedownthestepsyoutooktoworkouttheanswer.Workouttheansweragainusingadifferentmethod.Didyougetthesameanswer?If not,checkyourworking.

Key pointTo multiply two fractions, multiply their numerators and multiply their denominators.

of14

23

23

1 _ 4 of 2 _ 3 = 1 _ 6

Key pointSometimes you can rearrange fractions so they can be simplified before multiplying.

Rewrite the calculation with a fraction that can be simplified. 2 is a factor of 8 and 3 is a factor of 9.

Simplify the fractions before multiplying.

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Pi 3, Section 4.4

87

Exercise 4.51 Writedownthecommonfactorsof eachpairof numbers.

a 20and35 b 16and40 c 27and36 d 25and55

2 Workout

a 1_4×8 b 2_3×10 c 4_5×2_3 d 9__10×

1_3

3 Writeeachimproperfractionasamixednumber.

a 8_3 b 35__6 c 28__3

4 Completethesecalculationsforeachdiagram.a

14

14

14

14

1

1 ÷ 4 = u 1 ÷ u = 4 4 × u = 1

b

13

13

13

13

13

13

2

c

12

12

12

12

12

12

3

5 UseyouranswerstoQ4toworkouta howmanyquartersarein1 b howmanythirdsarein2c howmanyhalvesarein3 d howmanyquartersarein3.

6 Copyandcomplete.

a 4÷1_2 b 3÷1_5 c 2÷1_6 d 6÷1_4

7 Howmany1_4litrebottlescanbefilledfroma3litrecontainer?

8 Katiecuts4chocolatecakesintoeighths.Howmanyslicesarethere?

9 Reasoning Chriscuts3carrotcakesintoequalslices.Hehas30slices.Whatfractiondidhecuteachcakeinto?

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2 ÷ 6 = 1 _ 3 2 ÷ u = u u × u = u

3 ÷ u = u 3 ÷ u = u u × u = u

Q6a hint

How many halves are in 4?

Master

4.5 Dividing fractionsYou will learn to:• Dividebyfractions.

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Fluency How many• 2s are in 8• 4s are in 20• 1s are in 3• 5s are in 100?

explore Does division always make something smaller?

Check P93

extend P99

strengthen P95

test P103

Why learn thisReal-life measurements are not usually whole numbers. You need to be able to calculate with fractions too.


10 Writethereciprocalof eachfractionornumber.

a 2_3=3 __ u

b 3_4 c 11__4 d 15__2

e 2 f 4 g 1_3 h 1__10

Worked example Workout5÷2_3

5 ÷ 2 _ 3 = 5 _ 1 × 3 _ 2

= 15 __ 2

= 7 1 _ 2

11 Workout

a 6÷2_3 b 4÷3_4 c 10÷5_9 d 12÷3__10

e 18÷2_9 f 21÷7__10 g 15÷1_3 h 26÷2__5

12 Howmany3_4kgbagsof potatoescanbefilledfroma12kgsack?

13 STEM Anelectricianneedstocuta10mrollof cableintolengths5_6of ametre.Howmanylengthscanshecutfromtheroll?

14 Decideif thesestatementsaretrueorfalse.Giveexamplestohelpexplainyouranswers.a Thereciprocalof aproperfractionisanotherproperfraction.b Thereciprocalof animproperfractionisaproperfraction.c Whenyoumultiplytwofractionstheanswerisalwayslessthan1.d Whenyoumultiplytwoproperfractionstheanswerisalwayslessthan1.

e Whenyoumultiplyanintegerbyaproperfractiontheanswerismorethanthatinteger.

f Whenyoudivideanintegerbyaproperfractiontheanswerisgreaterthantheinteger.

15 Explore Doesdivisionalwaysmakesomethingsmaller?Choosesomesensiblenumberstohelpyouexplorethissituation.Thenusewhatyou’velearnedinthislessontohelpyouanswerthequestion.

16 Reflect FractioncalculationscanoftenbeshownusingbarmodelsasinQ4.Dothesediagramshelpyouunderstandhowtodividebyfractions?Explain.

Q10 Literacy hint

The reciprocal of a fraction is the ‘upside down’ fraction.

Q10e hint

The number 2 can be written as 2 _ 1 .

Key pointDividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is the ‘upside down’ fraction.

Multiply by the reciprocal of 2 _ 3 . 5 = 5 _ 1 .

Write as a mixed number in its simplest form.

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Pi 3, Section 4.5

89


a 25%of £1200 b 15%of £1200

c 30%of £1200 d 75%of £1200

2 Writethesescoresinorder,fromlowesttohighest.

6outof 10 13outof 20 14outof 25 29outof 50

3 Writedown6setsof equivalentfractions,decimalsandpercentagesfromthese.

120

13

35

34

45

19200.05

0.3

0.6

0.75

0.8

0.95

5%

33.3%

60%

75%

80%

95%

.

.

4 Writetheseamountsinorder,fromsmallesttolargest.

a 0.409 4_9 41% 11outof 20

b 0.67 66% 34outof 50 2_3

5 Reasoning Whichisbettervalue?

a 1_3off oradiscountof 30%

b Areductionof 15%orasavingof 1__10

c 20%off orpay4_5of thecost

6 Problem-solving Aswimmingteamhas20members.11membersaregirls.Whatpercentageof theswimmingteamareboys?

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topic links: Statistics, Proportion, Pie charts subject links: Cookery (Q7), Geography (Q13)

Master

4.6 Comparing proportionsYou will learn to:• Useacalculatortoworkoutpercentages• Compareproportions.

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FluencyWork out 20% and 5% of• 500• 360• 1800

exploreHow do calculators show 150%?

Check P93

extend P99

strengthen P95

test P103

Why learn this?Nutritional information on food packets can be given as a fraction, a percentage or as a decimal.


7 Reasoning / Problem-solving Hereissomenutritionalinformationfortwosimilarproducts.a Writetheproportionof fatinBrandAasapercentage

b Whichbrandhasahigherproportionof fat?c Writetheproportionof proteininBrandAasapercentage.

d Whichbrandhasahigherproportionof protein?e Whichbrandhasahigherproportionof carbohydrate?

8 Problem-solving Theratioof boystogirlsinaclassis11:14.Whatpercentageof theclassaregirls?

9 Problem-solving/Reasoning Thispiechartshowsthelanguagesstudiedbystudentsataschool.

Spanish

Languages being studied

French

German150° 120°

a 15peoplestudyFrench.Howmanypeopleweresurveyed?b Howmanypeoplestudy

i German ii Spanish?

10 Problem-solving /Reasoning Dave’sdrivingtheorytesthad20questionsinit.Hescored70%.Howmanyquestionsdidhegetwrong?

11 Problem-solving/Reasoning 120studentsvotedinaschoolcouncilelection.56peoplevotedforMatt,therestvotedforHassan.Whatpercentageof votesdidHassanwin,tothenearest1%?

12 Maxscored200outof apossible250inhismathstest.Whatwashisscoreasapercentage?

13 Real Thereare50statesintheUSA.27of themhavenocoast.Whatpercentageof thestateshavenocoast?

14 Explore Howdocalculatorsshow150%?Lookbackatthemathsyouhavelearnedinthislesson.Howcanyouuseittoanswerthisquestion?

15 Reflect Q10andQ11aretaggedasproblem‑solvingquestions.Thismeansthattherearealternativewaysof findingtheanswer.Whichmethodormethodsdidyouchoose?Explainwhy.

Brand A

Per 100 g

Protein 12.5 g

Carbohydrate 23 g

Fat 4.5 g

Brand B

Percentage content

Protein 15%

Carbohydrate 22.5%

Fat 5%

Q9 strategy hintUse the pie chart to work out the fraction of students who study French.

Q12 hint

200 out of 250 = 200 ___ 250

= 200 ÷ 250 e

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Pi 3, Section 4.6

91

Exercise 4.7: Percentage change1 Workout30%of

a 70 b 25 c 10 d 44 e 90

2 Writeeachimproperfractionas i amixednumber ii adecimal iii apercentage.

a 150___100 b 375___150 c 60__50 d 225___50

3 Increaseeachamountby20%.a £160

×210%

20%

£160

=

=

+ =

×2

b £3400

×210%

20%

£3400

=

=

+ =

×2

c £25000

4 Workouta £180increasedby20% b $2600increasedby15%c €2500increasedby30% d £4200increasedby25%e $3500increasedby5% f €4250increasedby50%

5 Decreaseeachamountby25%.a £240÷4=u

25% of £240 =u£240 – u=u

b £3200 c £24000

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Key pointAn increase of 20% means original amount (100%) + 20% of that amount = 120% of the amount.

Q5a hint

25% = 1 _ 4 so divide by 4.

Master

4.7 FINANCE: Percentage change You will learn to:• Workoutapercentageincreaseordecrease.

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Fluency Write 8 as a percentage of• 10• 16• 100• 200

explore What does ‘Up to 100% less sugar’ actually mean?

Check P93

strengthen P95

Why learn this? Understanding percentage change means you can make sure you don’t get overcharged or short-changed.

extend P99

test P103


6 Workouta £80decreasedby5% b $1200decreasedby20%c £680decreasedby10% d €910decreasedby1%e £8050decreasedby15% f $3400decreasedby30%

7 Finance AtravelcompanyadvertisestheseholidaysinJune.Albania £650Bulgaria £725Croatia £856Allthepricesgoupby12%inJuly.HowmuchiseachholidayinJuly?

8 Real / Finance a Anewspaperheadlinereads,‘Housepricesriseby100%inthelastdecade’.Whatdoesthatmean?b May’shousehasincreasedinvalueby100%sincesheboughtit.Itcosther£125000.Howmuchisitworthnow?

9 Finance a Acoatcosts£180.Itisreducedby30%inasale.Whatisitssaleprice?b Apairof jeanscost£70.Theyarereducedby45%inasale.Whatisthesaleprice?

10 Finance Workoutthecostof eachitemwhen20%VATisadded.a Acomputerthatcosts£720b Amealthatcosts£55c Ahaircutthatcosts£18

11 Finance / Problem-solving Janetneedsanewsofa.Whichshophasthecheaperprice?

Shop ASofa: £35020% off!

Shop BSofa: £340

15% off!

12 Reasoning Elisaves£10,whichis20%of hisallowance.Howmuchishisallowance?

13 Finance Asavingsaccountadvertises:

Earn 2% on your investment each year

Sueinvests£2000.a HowmuchinterestwillSuereceiveeachyear?b Suedoesn’tputanymoremoneyintotheaccount.HowmuchmoneywillSuehaveintheaccountintotalafter5years?

14 Reasoning Jamie’ssavingsaccountpays5%interesteachyear.Hereceives£10after1year.Howmuchmoneywasintheaccount?

15 Explore Whatdoes‘Upto100%lesssugar’actuallymean?Lookbackatthemathsyouhavelearnedinthislesson.Howcanyouuseittoanswerthisquestion?

16 Reflect Inthislessonsomequestionsareabout‘interest’.Explaininyourownwordswhatismeantby‘interest’andwhyitmightbeusefultounderstanditsmeaning.

Q8 hint

An increase of 100% means add on 100%.

Q10 Literacy hint

Vat (Value Added Tax) is a tax that is added to some goods and services before you buy them.

Q13 hint

The interest is only paid on the original investment.

Q14 hint

£10 is 5%How much is 10%?How much is 100%?

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Pi 3, Section 4.7

FINANCE

93

Master P79

4 Check up Log how you did on your Student Progression Chart.

Extend P99

Test P103

Strengthen P95ChECk

Equivalent proportions 1 Copy and complete the table showing equivalent fractions, decimals

and percentages.Write fractions in their simplest form.

Fraction Decimal Percentage

0.45

2 __ 25

5%

1 1 _ 2

0. 3 ˙

2 Write 0.82 as a fraction in its simplest form.

3 Write each fraction as a decimal.

a 3 _ 8 b 1 _ 9

4 Write < or > between each pair of fractions.

a 19 __ 25 u 11 __ 15 b 2 _ 3 u 7 __ 10

5 Write these values in order, smallest to largest.

11 __ 20 0.51 54.5% 8 out of 15

Fraction calculations 6 Work out

a 5 _ 9 + 1 _ 6 b 9 __ 10 – 3 _ 4

7 Work out the sum of 1 _ 4 , 5 _ 6 , and 2 _ 3 . Write your answer as a mixed

number.

8 Work out each calculation. Write your answers in their simplest form.

a 3 _ 4 × 3 _ 5 b 2 _ 9 × 3 _ 7 c ( 2 _ 5 ) 2

9 Work out

a 4 1 _ 5 + 1 7 __ 10 b 1 3 _ 4 + 2 3 _ 5 c 3 9 __ 10 – 1 4 _ 5 d 2 2 _ 3 – 1 3 __ 10

10 Chantel needs 5 lengths of ribbon. Each length is 3 _ 4 metres. She says, ‘I know I will need less than 5 metres altogether’. How does she know?

11 Work out 6 __ 13 × 2 _ 3

12 Work out

a 7 ÷ 1 _ 2 b 4 ÷ 1 _ 5 c 8 ÷ 2 _ 3

13 Write the reciprocal of

a 6 _ 7 b 7 c 2 _ 3


Percentages14 The ratio of boys to girls in a class is 9 : 11. What percentage of the

class are boys?

15 A football club has 10 male players and 15 female players. What percentage of the club area male b female?

16 Work outa £460 increased by 25% b $4500 increased by 15%.

17 Anna has to pay 20% tax on a laptop. The laptop costs £450 before tax. How much does Anna pay in total?

18 Brian invests £5000 with yearly interest paid at 1%. How much interest will he earn after 1 year?

19 A flat increased in value by 150%. It was worth £200 000. What is its new value?

20 Work out a £30 reduced by 25% b £200 reduced by 5%

21 Steve’s car has gone down in value by 15%. He paid £8600. What is it worth now?

22 20% of an amount is $25. What is the amount?

23 How sure are you of your answers? Were you mostly

Just guessing Feeling doubtful Confident

What next? Use your results to decide whether to strengthen or extend your learning.

Challenge24 Jarred draws a square.

He colours in half red. He colours in half of the rest orange.

He continues like this, colouring in half of what’s left using the colours of the rainbow: red, orange, yellow, green, blue, indigo, violet.a What fraction of the square will be coloured indigo?b What fraction will be coloured violet?c Freya tries a similar experiment, colouring in 1 _ 3 of the shape each

time. Explore the different fractions in Freya’s shape.

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Master P79

Test P103

You will:• Strengthen your understanding with practice.

Extend P99

Check P93

4 Strengthen

STRENGThEN

Equivalent proportions1 Write each of these as a decimal number.

a 43 ÷ 100 b 17 ÷ 100c 32 ÷ 100 d 98 ÷ 100e 52% f 65%

2 Write each decimal as a i percentage ii fraction in its simplest form.

a 0.13 b 0.32 c 0.68

3 Write each fraction as a percentage.

a 3 _ 4 b 3 __ 50 c 7 __ 20 d 17 __ 20 e 19 __ 25

4 Match up equivalent fractions, decimals and percentages.

25

1.4 140% 1

14

1.25125% 1

45

0.880%

15

2.2

220% 2

5 Write each decimal as a fraction. Simplify where possible.

a 0.123 = 123 ___ u

b 0.763

c 0.442 d 0.125 e 0.988 f 0.375 g 0.128

6 Rewrite each recurring decimal using dot notation. a 0.666 6… b 1.333 3…c 2.151 515 … d 3.567 567 567 …

7 Write these decimals in order, from smallest to largest.a 0.3 0. 3 ˙ 0.329 0.32 b 0. 4 ˙ 0.45 0.439 0.4c 0. 5 ˙ 0.549 0.5 0.56d 0.665 0.68 0. 6 ˙ 0.6

8 Use a written method to work out each division.

a 235 ÷ 5 b 135 ÷ 9 c 856 ÷ 8

9 Use a written method to write each fraction as a decimal.

a 7 _ 8 b 3 _ 8 c 5 __ 16 d 3 __ 16

Q1e hint

52% is the same as 52 out of 100 = 52 ÷ 100

Q2b hint

0.32 = 32

___ 100

Simplify the fraction.

Q3 hint

Write an equivalent fraction with denominator 100.

Q6 Literacy hintA dot above a digit shows it recurs.6. 4 ˙ = 6.444 4… A dot above the beginning and end of a sequence shows the whole sequence recurs.9. 3 ˙ 9 4 ˙ = 9.394 394 394 394 …

Q7a hint

Which is larger: 0.3 or 0.333 3….?

Q8a hint

235 _ 5)

Q9a hint

8 7. )0.uuu

70 0 0


10 Which of these fractions will give a recurring decimal?

45

23

19

311

916

37

56

12

78

310

11 Write < or > between each pair of fractions.

a 1 _ 3 u 3 __ 10 b 3 _ 5 u 2 _ 3 c 4 _ 7 u 5 _ 9 d 8 _ 9 u 9 __ 11

12 Match each proportion to a bar. Use them to write the proportions in order, from smallest to largest.

A 0.7 B 0.5 C 60% D 2 _ 3 E 3 _ 4

i

ii

iii

iv

v

13 Write these proportions in ascending order.

a 4 _ 5 0.4 30% 0.6 2 _ 3 b 95% 5 _ 6 0.9 3 _ 4 0.85

Fraction calculations 1 Write each improper fraction as a mixed number.

a 11 __ 3 b 22

__ 5

c 15 __ 4 d 24 __ 9

2 Write each mixed number as an improper fraction.

a 1 4 _ 5 b 2 3 _ 4

c 2 8 _ 9 d 3 4 _ 7

3 Work out each calculation. Write your answer as a mixed number.

a 2 _ 8 + 5 _ 8 + 7 _ 8 = u

__ 8 = 1

u __

8 b 4 _ 9 + 7 _ 9 + 8 _ 9

c 4 _ 5 + 2 _ 5 + 3 _ 5 d 5 __ 12 + 7 __ 12 + 11 __ 12

4 Work out the missing number.

a 5 _ 9 + 3 _ 9 + u = 1 b 1 – 2 _ 9 – 4 _ 9 = u

c 1 __ 12 + 5 __ 12 + u = 1 d 1 – 3 _ 7 – 2 _ 7 = u

e 3 _ 5 + 1 __ 10 + u = 1 f 7 __ 12 + 1 _ 6 + u = 1

g 1– 1 _ 5 – 3 __ 10 = u h 1 – 5 _ 8 – 1 _ 4 = u

Q11 hint

Write each fraction as a decimal.

Q13 Literacy hintAscending means getting larger.

Q1a hint

5 1

113

33 5 13

3 5 133

23

Q2a hint

1 = 5 __

5 5 __

5 +

4 __

5 =

u ___

5

Q4a hint

How many ninths equal 1?

97

5 Problem-solving What fraction should the third sector on this pie chart be labelled?

6 Work out

a 2 1 _ 3 + 1 4 _ 5 = 2 + 1 + 1 _ 3 + 4 _ 5 b 4 5 _ 6 + 1 2 _ 5

= u + u

__ u

c 3 4 _ 5 – 1 3 _ 4 = (3 – 1) + ( 4 _ 5 – 3 _ 4 ) d 5 2 _ 3 – 3 1 _ 4

= u + u

__ u

7 Work out

a 2 _ 3 of 1 _ 2 = 2 × 1 ____ 3 × 2

= u

__ u

b 1 _ 4 of 4 _ 5

c 1 _ 3 × 3 _ 7 d 1 _ 4 × 3 _ 4

8 Work out

a 3 _ 4 × 5 _ 9 b 3 _ 5 × 3 _ 8 c 5 _ 9 × 2 _ 5

9 Work out

a 3 ÷ 1 _ 4 = 3 _ 1 × 4 _ 1 = u b 3 ÷ 3 _ 4 = 3 _ 1 × 4 _ 3 = u

c 6 ÷ 2 _ 3 d 6 ÷ 3 _ 5

Percentages1 The price of a computer has increased by 15%. It was originally £450.

What is the new price?

2 Reasoning Frankie works out £520 reduced by 20% like this:

10% of £520 = £52

80% of £520 = £416x8 x8

Rob works it out like this:

10% of £520 = £52

20% of £520 = £104

£520 – £104 = £416

×2×2

Whose method do you prefer? Why?

3 The value of a car has gone down by 25%. It was originally worth £8000. What is the new value?

Q5 Strategy hintWrite the question as a subtraction calculation.

12

13

Q7a hint

Finding a fraction of an amount is the same as multiplying by the fraction.

of23

12

13

Q9a hint

How many quarters in 3?

Q1 hint

10% of 450 = u 5% of 450 = u15% of 450 = u + u = u 450 + u = u

Q2 hint

100% − 20% = 80%


4 A holiday costs £1200. Booking before the end of the month saves 15%. How much is the holiday with the saving?

5 Reasoning / Problem-solving

a 10% of a number is 25. What is the number?

b 25% of a number is 50. What is the number?

c 30% of a number is 24. What is the number?

d 40% of a number is 320. What is the number?

Enrichment1 Draw a pie chart to show how you spent the 24 hours in a weekday

during term-time. What fraction or percentage of your time is spenta sleepingb in lessonsc eating/washing/getting readyd socialising/entertainment/TVe homework?

2 How would your pie chart from Q1 be different for a Saturday in school holidays?

3 Reflect List these tasks in order from easiest to hardest. A Adding and subtracting fractions B Multiplying fractionsC Dividing fractionsD Adding and subtracting mixed numbersE Multiplying mixed numbersF Dividing mixed numbersLook at the first task in your list (easiest). What made it easiest?Look at the two tasks at the bottom of your list (hardest). What made them hardest?Write a hint to help you with the two tasks you found hardest.

Q5c Strategy hint30% = 24

43 4310% = u

310 310100% = u

Q1 hint

Number of degrees for each hour = 360° ÷ 24 hours.

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Q3 hint

List the letters of the tasks in order. You don’t need to write out the descriptions.

99

You will:• Extend your understanding with problem-solving.

4 Extend

Test P103EXTENDStrengthen

P95Check

P93Master

P79

1 Write each percentage as a decimal. a 12.5% b 58.3% c 5.4%d 2.9% e 14.75% f 3.25%

2 Write each decimal as a percentage.a 0.185 b 0.529 c 0.095d 0.1234 e 0.0987 f 0.0037

3 Write each terminating decimal as a fraction in the simplest form.a 0.18 b 0.99 c 0.175 d 0.025e 1.65 f 4.848 g 3.05 h 6.02

4 Work out the first amount as a percentage of the second amount. a 80p out of £2 b 750 ml out of 3 litresc 40 minutes out of 2 hours d 250 cm out of 4 metrese 3.5 kg out of 5 kg f 2.4 km out of 6 km

5 98 out of 180 members of a tennis club are male.a What fraction of the club are female?b What percentage of the club are male, to the nearest 1%?

6 Real 20 out of 196 countries in the world have Spanish as their official language. What percentage of countries is that, to the nearest 1%?

7 Reasoning Sort these fractions into terminating or non-terminating decimals.

1 _ 2 1 _ 3 1 _ 4 1 _ 5 1 _ 6 1 _ 7 1 _ 8 1 _ 9 1 __ 10 1 __ 11 1 __ 12

8 Write these fractions in order from largest to smallest.

a 3 _ 4 17 __ 20 2 _ 3 3 _ 5 b 2 _ 5 3 __ 10 7 __ 20 11 __ 25

9 Reasoning Write each fraction as a decimal to show why each statement is true.

a 1 _ 3 > 3 __ 10 b 2 _ 3 > 3 _ 5 c 4 _ 9 < 1 _ 2 d 3 _ 4 < 7 _ 9

10 Write each time as a mixed number.

a 1 hour 45 minutes = 1 u

___ u

hours b 1 hour 10 minutes

c 5 hours 40 minutes d 2 hours 15 minutes

e 3 hours 55 minutes f 2 hours 5 minutes

g 1 hour 25 minutes h 20 minutes

i 35 minutes j 50 minutes

11 Write each of the times in Q10 as a decimal number of hours.

Q4 hint

Make sure the units are the same.

Q7 Strategy hintUse division if you are not sure.

Q8 Strategy hintWrite each fraction as a decimal to compare them.

Q10a hint

45 minutes = 45

__ 60 of an hourSimplify the fraction.


12 Work out each calculation. Give your answer in hours and minutes.

a 2 hours 30 minutes × 5 = u

__ u

hours × 5 = ___ hours ___ minutes

b 4 × 1 hour 45 minutes c 3 hours ÷ 30 minutes

13 Modelling / Problem-solving A film lasts 2 hours 15 minutes. A cinema plays it continuously from 13:00 to 22:00. How many times is it played?

14 A bus journey to the city lasts 1 hour 15 minutes each way. Raghev travels by bus into the city and back three times a week. What is the total time Raghev spends travelling by bus to the city each week?

15 Match pairs of times.

A 1 hour 6 minutes i 2.9 hours

B 2 hours 3 minutes ii 1.35 hours

C 2 hours 54 minutes iii 1.1 hours

D 1 hour 12 minutes iv 2.8 hours

E 2 hours 48 minutes v 2.05 hours

F 1 hour 21 minutes vi 1.2 hours

16 Real 53 million people were recorded in the 2011 census for England. Of these, approximately 6 million people were under 10 years old. What percentage were under 10?

17 A restaurant bill costs £100 plus taxes. The final bill including tax is £110. What percentage of the restaurant bill is tax?

18 Work out the percentage increase. The first one has been started for you. a £80 increased to £100

Difference: 100 – 80 = 20

Difference ____________ Original amount

= 20 ___ 80

= 1 _ 4 = u%

Increase of u% Check: £80 + 25% = u

b £120 increased to £144

Difference: 144 – 120 = uc £1500 to £1650 d £360 to £405 e $420 to $453.60

19 Before tax a bike costs £490. With tax it is £578.20. What percentage was the tax?

20 A school has increased in size from 2200 pupils to 2464 pupils. By what percentage has the school size increased?

21 The cost a bag of bananas has risen from £1.40 to £1.47. What percentage increase is this?

Q18 hint

Percentage = amount of ÷ original change increase amount

101

22 Work out the percentage decrease. The first one has been started for you.a £80 decreased to £68

Difference: 80 – 68 = 12

Difference ____________ Original amount

= 12 ___ 80

= 0.15

= u% Decrease of u%

b £120 decreased to £96

Difference: 120 – 96 = uc £1500 to £1125

d £360 to £342

e $250 to $218.75

23 Real A ski helmet is reduced from £150 to £97.50. What percentage is it reduced by?

24 Real A jacket costs £120. There is £15 off. What percentage reduction is this?

25 Work out

a 3 2 _ 5 + 2 1 _ 3 + 2 7 __ 10 b 4 5 _ 6 – 2 2 _ 3 – 1 2 _ 9 c 1 _ 2 × 3 _ 4 × 5 _ 6

d 2 _ 3 × 6 _ 7 × 3 _ 4 e (3 ÷ 3 _ 4 ) × 1 _ 3 f 3 ÷ ( 3 _ 4 × 1 _ 3 ) g (6 ÷ 2 _ 3 ) × 1 _ 4 h 6 ÷ ( 2 _ 3 × 1 _ 4 )

26 Use division to write each fraction as a decimal.

a 7 __ 11 b 4 __ 15 c 17 __ 24

27 Write the reciprocals of each number. Give your answers in their simplest form.

a 7 __ 11 b 4 __ 15 c 3 5 _ 6

28 Work out the sum of 3 2 _ 9 , 1.8 and 4 2 _ 3

29 How many 3 _ 4 litres bottles can be filled from a 9-litre container?

30 Problem-solving Graham, Andrea and Carly are playing a missing-number game.Graham says, ‘35% of my number is 70’.Andrea says, ‘15% of my number 75’. Carly says, ‘30% of my number is 60’.What is the sum of all three of their numbers?

Investigation Reasoning / Modelling / Problem-solving1 Write the next four terms in each of these fraction sequences.

a 1 _ 2 , 1

_ 3 , 1

_ 4 , 1

_ 5 , … b 1 _ 2 , 2

_ 3 , 3

_ 4 , 4

_ 5 , … c 9 __ 10 , 8

_ 9 , 7

_ 8 , …

2 Write each term in the sequence as a decimal number.3 Bhavika says, ‘Each sequence is getting closer and closer to a number’.

For each sequence, write down which number it is getting closer to.Discussion Do you think the sequence will reach that number? Explain

Q22 Strategy hintCheck your answer by working out the decrease.

Q27c hint

Write mixed numbers as improper fractions first.


31 a Match each statement to the percentage it gives of the original amount.

A An increase of 30%

B A decrease of 10%

C Interest of 18%

D Interest gain of 1%

E Saving of 30%

F 18% reduction

G 1% less

i 70%

ii 82%

iii 130%

iv 99%

v 90%

vi 118%

vii 101%

b Write each percentage in part a as a decimal.

32 Work out these percentages of amounts using a decimal multiplier. a 80% of 2756 = 0.8 × 2756 = u b 5% of 650 = 0.05 × u = c 72% of 3675 d 90% of £148e 7% of £1240 f 125% of £32 000

33 Use a decimal multiplier to work outa £120 increased by 30% b £1500 increased by 15% c $4800 increased by 5% d €2240 increased by 12.5%.

34 Use a decimal multiplier to work outa £1240 decreased by 10% b £3500 decreased by 15% c $2800 decreased by 30% d $7200 decreased by 22%e €1242 decreased by 5% f €1250 decreased by 7.5%.

35 Problem-solving / Finance An insurance company offers two ways of paying.

Only £22 per monthfor a year!

ONE-OFF PAYMENT OF £240.

a How much more expensive, overall, is the pay monthly option?

b What percentage of the yearly payment is added to the monthly option?

36 Reflect Write down three new skills you have learned in this unit. For each one of these, choose a question that used this new skill.

key pointYou can multiply an amount by a decimal multiplier to work out percentage change.

Q33a hint

An increase of 30% = 100% + 30% = 130% = u.u

Q34a hint

A decrease of 10% = 100% – 10% = 90% = u.u

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4 Unit test Log how you did on your Student Progression Chart.

TESTExtend P99

Strengthen P95

Check P93

Master P79

1 Write each decimal number as a percentage.a 0.15 b 0.2c 0.01 d 3.4

2 Write each decimal number as a fraction or mixed number in its simplest form.a 0.08 b 1.25

3 Write 13 __ 20 as a a percentage b a decimal.

4 Write these fractions in order, smallest to largest.

7 __ 10 6 _ 5 11 __ 20 12 __ 25 7 __ 10 6 _ 5 11 __ 20 12 __ 25

5 Write 4 __ 15 as a decimal.

6 Work out

a 3 _ 5 + 1 _ 6 b 8 _ 9 – 2 _ 3

7 Work out 4 _ 7 + 5 _ 7 + 6 _ 7 . Give your answer as a mixed number in its simplest form.

8 Work out

a 4 4 _ 5 – 1 3 __ 10 b 2 2 _ 3 + 1 4 _ 5

9 In a triathlon, Rich swims 1 3 _ 4 km, cycles 15 7 __ 10 km and runs 8 4 _ 5 km. What is the total distance covered by the triathlon?

10 The ratio of boys to girls in a class is 13 : 12. a What fraction of the class are girls? b What percentage of the class are girls?

11 Increase $2640 by 5%.

12 Work out 4 ÷ 1 _ 6

13 Work out each calculation. Give your answers in their simplest form.

a 4 __ 15 × 3 _ 8 b 1 _ 2 × 3 _ 5 × 5 _ 9

c 3 ÷ 3 _ 4

14 Write the reciprocal of

a 11 __ 12 b 3

c 1 __ 10 d 12 __ 5

15 A house increased in value by 200%. It was originally worth £215 000. What is its new value?


16 Reduce £470 by 6%.

17 A restaurant bill says £60.50 + 14% service charge. How much is the total bill, with service charge?

18 25% of a number is 14. What is the number?

19 A truck has gone down in value from £50 000 to £42 500. By what percentage has the value reduced?

20 Write 2 hours 15 minutes as a a fraction of an hourb a decimal proportion of an hour.

21 Copy and complete, writing < or >. 4 _ 5 u 7 _ 9 .

22 Write the decimal multiplier for working out aa 90% increase b 3% increasec 40% decrease d 2% decrease

Challenge23 Reasoning Use these number cards to make 2 different proper

fractions, for example 2 _ 5 and 1 _ 3 .

2 5 1 3

a Work out the sum of your two fractionsb Work out the difference between your two fractions.c Do you think the sum is as large as possible? If not, make two

different fractions using the numbers. d Do you think the difference is as small as possible? If not, make

two different fractions using the numbers.e How can you make sure the sum is as large as possible?f How can you make sure the difference is as small as possible?g Try with 4 different numbers.

24 Reflect Look back at this test. Which questions took the shortest time to answer and which took the longest? Why do you think you could answer some questions more quickly than others? R

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